In a statement, A.M.I. called the settlement an “amicable resolution” and said that under the new agreement it would devote an upcoming magazine cover and feature article to Ms. McDougal, and would run several of her fitness columns in its publications.

“It’s a total win,” Mr. Stris said in an interview. “We got everything we were fighting for — she got out of the contract, gets the life rights back and owes A.M.I. nothing more.”

In a separate interview, Ms. McDougal expressed elation about the end of her “wild ride,” and said she currently had no plans to sell the rights to her story to a new buyer. “It’s one step at a time for me,” she said. “Today, I’m doing my victory dance.”

Her lawsuit said that A.M.I., whose chairman, David J. Pecker, is a friend of President Trump’s, misled her into signing the contract. It also claimed that Mr. Cohen had inappropriately intervened in the deal. A.M.I. had denied misleading her.

The deal and the extent of Mr. Cohen’s role in it are the subjects of a wide-ranging federal corruption investigation that is, in part, looking into his efforts to protect Mr. Trump’s presidential prospects in 2016.

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A.M.I. indicated earlier this month that it would fight Ms. McDougal, asking the Los Angeles Superior Court to dismiss her lawsuit.

But that was roughly a week before federal investigators obtained email communications, audio recordings and other documentation from Mr. Cohen during their raid of his office, home and hotel room. Those materials included information about A.M.I. and the McDougal suit, people involved in the case said.

The suit also claimed that Mr. Cohen had been secretly involved in the talks between A.M.I. and Ms. McDougal’s lawyer at the time, Keith M. Davidson — who emailed Mr. Cohen at the end of the negotiations. A spokesman for Mr. Davidson has said the lawyer “fulfilled his obligations and zealously advocated for Ms. McDougal.”

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A.M.I. also spoke with Mr. Cohen about Ms. McDougal, though it says it did so only as part of its reporting process.

Mr. Stris said that before reaching the settlement, he was prepared to answer A.M.I.’s motion to dismiss Ms. McDougal’s case with a request for a limited version of pretrial discovery. The move, compelling both sides to share emails and other records, could have provided information that would not be available through the material the F.B.I. seized from Mr. Cohen. An A.M.I. lawyer, Cameron Stracher, said that he doubted the request would have succeeded, and that avoiding discovery was not a motivating factor in the settlement.

The agreement precludes any of that from happening, at least in Ms. McDougal’s civil case against A.M.I., though Mr. Stris said he expected federal investigators to eventually secure everything they needed to fully vet the process behind the deal. “I have tremendous confidence in the men and women of the Southern District of New York,” he said, referring to the federal prosecutors investigating Mr. Cohen.

Mr. Stris said he did not rule out taking legal action if more came to light about potentially inappropriate contact between Mr. Cohen and Mr. Davidson during the course of negotiating her deal. The settlement does not bar Ms. McDougal from claims against either of the men in the future. For now, Ms. McDougal said, “I need to relax and get my life back and de-stress.”

The initial deal prohibited Ms. McDougal from speaking about her alleged affair, but A.M.I. amended the contract after the election to allow her to answer “legitimate press inquiries,” in response to her complaints that the agreement was overly restrictive. In recent months, she has spoken to The New Yorker and Anderson Cooper of CNN.

Separate from the federal investigation into Mr. Cohen, A.M.I. is facing a complaint at the Federal Election Commission that its $150,000 payment to Ms. McDougal was an illegal campaign expenditure. The publisher has denied this and says it acts solely as a news organization with a First Amendment right to run stories — or not run them — as it chooses.

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The Internal Revenue Service announced late Tuesday that it would let taxpayers submit tax returns without penalty through Wednesday, after a long day of technical problems that fueled confusion about what is already one of Americans’ most frustrating interactions with their government.

A computer glitch at the IRS knocked offline the agency’s ability to process many tax returns filed electronically, a stunning breakdown that left agency officials flummoxed and millions of Americans bewildered. Senior government officials were at a loss to explain what happened, even as close to 5 million Americans were expected to try to file their taxes before the midnight deadline.

IRS officials did not specify what went wrong, saying only that they would undertake a “hard reboot” of their systems. By late Tuesday, the IRS said that its systems were back online and that taxpayers could proceed to file returns through the end of Wednesday. Taxes had been due on Tuesday. (That was two days later than the usual due date, April 15, which fell on a Sunday. Monday was Emancipation Day, a holiday in the District.)

“This is the busiest tax day of the year, and the IRS apologizes for the inconvenience this system issue caused for taxpayers,” said the agency’s acting commissioner, David Kautter.

An IRS spokeswoman said that “all indications point to this being hardware-related. We’re aware of no other external issues.”

The episode recalled other high-profile government technology breakdowns, such as the challenging launch of the Affordable Care Act marketplaces in 2013, and raised fresh questions about whether the IRS, which has long complained about antiquated computer systems, is prepared for the massive overhaul mandated by last year’s sweeping tax law.

Each year, millions of Americans are required to file tax returns by mid-April for money they earned the prior year. The process can be a financially and administratively painful one, but the IRS is supposed to have sophisticated computer systems that can handle millions of last-minute filers. Between 1 a.m. and 3 a.m. Tuesday, officials discovered that those systems had faltered.

It wasn’t immediately clear how many people were affected or could take advantage of the one-day delay in the filing deadline, but IRS officials said taxpayers wouldn’t have to do anything special to take advantage of the postponement. Many filers who use online tax preparation software, such as TurboTax or HR Block, or pay their taxes directly to the IRS online were affected. The vast majority of tax preparers, such as accountants, are required to file taxes electronically.

The IRS is often at the center of political fights in Washington, with Democrats calling for more funding so the agency can do its job while Republicans have worked for years to pare it back, before agreeing this year to fund the agency at higher levels.

The IRS has faced steady budget cuts for nearly a decade, with its staff size falling by about 18,000 employees from 2010 to 2017 and a recent report showing that it can answer only about 60 percent of calls from tax filers.

The agency is taking steps to implement changes required by a sprawling overhaul of the tax code that Republicans passed in December. It has been working with businesses to make sure they are withholding the correct amounts from employees’ paychecks, as well as rolling out online tools that workers can use to ensure their employers’ calculations are correct.

“The IRS is highly vulnerable to IT breakdowns and cyberattacks,” said Pete Sepp of the National Taxpayers Union, a nonpartisan group that has pushed for changes at the IRS.

Members of Congress expressed frustration with the agency’s performance.

“Unfortunately, it’s another example where they’re not capable of dealing with the volume,” said Sen. Rob Portman (R-Ohio), who has called for reforms at the IRS partly because of the agency’s technological shortcomings. 

“This is game day for the IRS, and it seems the IRS can’t get out of the locker room,” said Rep. Greg Gianforte (R-Mont.).

Sen. Ron Wyden (D-Ore.) noted that the agency’s budget has been repeatedly cut in recent years and said this could have contributed to the problems.

“While we don’t yet know what has caused this systems failure, the lack of Republican funding for the IRS to serve taxpayers will only compound the issue. Americans should not be punished for being unable to file their tax returns or pay their tax bills today,” said Wyden, the top Democrat on the Senate Finance Committee, which oversees the IRS.

IRS officials were surprised by the breakdown.

“On my way over here this morning, I was told a number of systems are down at the moment,” Kautter told lawmakers at an oversight hearing Tuesday morning. “We are working to resolve the issue, and taxpayers should continue to file as they normally would.”

“If we can’t solve it today, we’ll figure out a solution,” Kautter said. “Taxpayers would not be penalized because of a technical problem the IRS is having.”

Tuesday’s outage caught at least one White House official off guard, too. Larry Kudlow, President Trump’s top economic adviser, appeared not to know about the problems when asked about them shortly after noon Tuesday.

“The IRS is crashing?” he said, repeating a reporter’s question. “It sounds horrible. It sounds really bad. Hope it gets fixed.”

A spokeswoman for Intuit, the company that owns the TurboTax software, said Tuesday before the delay was announced that taxpayers should continue as usual.

“Taxpayers should go ahead and continue to prepare and file their taxes as normal with Turbo­Tax,” said spokeswoman Ashley McMahon. “TurboTax has uninterrupted service and is available and accepting e-filed returns. We will hold returns until the IRS is ready to begin accepting them again.”

The IRS has more than 60 different IT systems for managing the cases of individual taxpayers, according to a report submitted to Congress by an internal IRS watchdog. Many of them have not been updated in decades, and two of them are nearly six decades old — the oldest anywhere in the federal government, the report said.

In testimony in October, two senior IRS officials warned that the agency’s systems were at high risk.

“We are concerned that the potential for a catastrophic system failure is increasing as our infrastructure continues to age. Thus, replacing this aging IT infrastructure is a high priority for the IRS,” wrote Jeffrey Tribiano, deputy commissioner for operations support, and Silvana Gina Garza, chief information officer, in prepared testimony.

In his testimony before Congress on Tuesday, Kautter said the IRS had prioritized the core filing system in its technology spending.

For several hours Tuesday, an erroneous page linked to in the IRS’s online payment section described a “Planned Outage: April 17, 2018 — December 31, 9999.”

While Republicans previously favored scaling back the IRS, the party more recently supported efforts to better support the agency. Congress approved $320 million in short-term funding to help the IRS implement the new tax law as part of the massive budget deal passed in March, but many lawmakers say more money is needed.

The House is scheduled to vote this week on a bipartisan bill making major changes to the agency, including beefing up free tax advisory programs for the poor and giving taxpayers several new rights and protections.

Erica Werner, Ellen Nakashima and Anne Gearan contributed to this report.

Tyler Q. Houlton, a spokesman for the Department of Homeland Security, said the ruling would undercut “efforts to remove aliens convicted of certain violent crimes, including sexual assault, kidnapping and burglary, from the United States.”

“By preventing the federal government from removing known criminal aliens,” he said, “it allows our nation to be a safe haven for criminals and makes us more vulnerable as a result.”

The case, Sessions v. Dimaya, No. 15-1498, was first argued in January 2017 before an eight-member court left short-handed by the death of Justice Antonin Scalia. The justices deadlocked 4 to 4, and the case was reargued in October after Justice Gorsuch joined the court.

The case concerned James Dimaya, a native of the Philippines who became a lawful permanent resident in 1992, when he was 13. In 2007 and 2009, he was convicted of residential burglary.

The government sought to deport him on the theory that he had committed an “aggravated felony,” which the immigration law defined to include any offense “that, by its nature, involves a substantial risk that physical force against the person or property of another may be used in the course of committing the offense.”

In 2015, in Johnson v. United States, the Supreme Court ruled that a similar criminal law was unconstitutionally vague. Justice Elena Kagan, writing for the majority in Tuesday’s case, said the reasoning in the Johnson case also doomed the challenged provision of the immigration law.

She quoted at length from Justice Scalia’s majority opinion in Johnson, which said courts could not tell which crimes Congress had meant to punish.

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“We can as well repeat here what we asked in Johnson,” Justice Kagan wrote, paraphrasing Justice Scalia. “How does one go about divining the conduct entailed in a crime’s ordinary case? Statistical analyses? Surveys? Experts? Google? Gut instinct?”

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She added that lower courts had been unable to apply the immigration law consistently.

“Does car burglary qualify as a violent felony?” she asked. “Some courts say yes, another says no. What of statutory rape? Once again, the circuits part ways. How about evading arrest? The decisions point in different directions. Residential trespass? The same is true.”

Justices Ruth Bader Ginsburg, Stephen G. Breyer and Sonia Sotomayor joined all of Justice Kagan’s opinion, and Justice Gorsuch most of it.

When the Johnson case was before the Supreme Court, the government warned that a ruling striking down the law at issue there would make the immigration law “equally susceptible” to constitutional attack.

Both laws, the government said then, required courts to identify features of a hypothetical typical offense and then to judge the risk of violence arising from them.

But when Mr. Dimaya’s case reached the Supreme Court, the government said there were significant differences between the two laws, focusing on minor variations in their wording. In dissent, Chief Justice John G. Roberts Jr., joined by Justices Anthony M. Kennedy, Clarence Thomas and Samuel A. Alito Jr., made a similar point.

Justice Kagan responded that some of Chief Justice Roberts’s analysis was “slicing the baloney mighty thin.”

The government also argued that the two laws should be treated differently because one concerned crimes and the other immigration, which is a civil matter.

In its brief, the government said civil laws are almost never so vague as to violate the Constitution. “Although the court has on occasion tested civil provisions for vagueness,” the brief said, “it has struck down those provisions under the due process clause because they were so unintelligible as to effectively supply no standard at all.”

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Justice Kagan disagreed. “This court’s precedent forecloses that argument,” she wrote, “because we long ago held that the most exacting vagueness standard should apply in removal cases.”

A 1951 Supreme Court decision, Jordan v. De George, indicated that both criminal and immigration laws should be tested against the same constitutional standard for vagueness “in view of the grave nature of deportation.”

Near the end of her opinion, Justice Kagan again quoted Justice Scalia. “Insanity,” he wrote in a 2011 dissent, “is doing the same thing over and over again, but expecting different results.”

Justice Kagan said it was time to heed that advice. “We abandoned that lunatic practice in Johnson,” she wrote, “and see no reason to start it again.”

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Senate Republicans are escalating their attacks on West Virginia Senate GOP candidate Don Blankenship, increasingly worried that the coal baron and ex-prisoner will blow a winnable race against Democratic Sen. Joe Manchin.

Republicans see West Virginia as a prime pickup opportunity in November, given President Donald Trump’s huge popularity there. But they say the multimillionaire Blankenship, running in a tight three-way primary against Rep. Evan Jenkins (R-W.Va.) and state Attorney General Patrick Morrisey, is indefensible as a candidate after serving a year in prison for conspiring to violate mine safety violations. Twenty-nine miners died at his company’s Upper Big Branch mine in 2010.

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“Wasn’t he convicted of a crime?” Senate Majority Whip John Cornyn of Texas said in an interview Tuesday. “That sort of background doesn’t lend itself to public office, in my view. Being convicted of a crime is a real liability.”

As the race gets closer, alarms are going off in Senate leadership suites. At a Monday evening Senate Republican leadership meeting, National Republican Senatorial Committee Chairman Cory Gardner (R-Colo.) focused on Blankenship’s weakness, explaining to other Republicans that the former coal magnate would be difficult for the party to defend in the state, according to attendees.

He specifically referenced accusations that Blankenship’s companies contaminated drinking water. A case involving the issue was settled with West Virginia residents, though Massey Energy, which Blankenship led, admitted no wrongdoing.

“That’s not popular in West Virginia,” Gardner told leaders gathered in McConnell’s office, referring to the contamination claims.

The situation between Blankenship and national Republicans is growing more fraught by the day. Last week, a super PAC apparently controlled by national interests blasted Blankenship with the allegations about contaminating the state’s drinking water at the same time he built a private water supply for his home. Blankenship hit back Monday in a scathing statement likening Senate Majority Leader Mitch McConnell to the Russians, interfering in an election outside his jurisdiction.

Cornyn and other top Republican senators continued the tit-for-tat this week. And the famously strategic McConnell wryly claimed he doesn’t “pay a lot of attention to these primaries,” before throwing a jab at Blankenship.

“We’ll wait to see who the nominee is and get behind a Republican candidate. And hopefully it will be one who is actually electable,” McConnell said Tuesday, which he later called a “subtle hint” about his preference in the primary.

Meanwhile, Jenkins and Morrisey are beating up each other ahead of the May 8 primary. Morrisey released an ad on Tuesday attacking Jenkins’ “liberal Democrat record,” while Jenkins has focused on Morrisey receiving campaign money from the pharmaceutical industry, an attack that resonates in a state wracked by opioid addiction.

Officially, Senate Republicans aren’t intervening in the race out of fear it would aid Blankenship’s bid to portray himself as an outsider taking on the corrupt establishment. But privately, Blankenship is a hot topic among GOP leaders.

“It’s kind of wild out there. I don’t think he will [win]. I certainly hope one of the other candidates does, because I think he would be obviously a challenged candidate going into the fall,” said Sen. John Thune of South Dakota, the No. 3 GOP leader.

On Monday, Sen. Shelley Moore Capito (R-W.Va.), a McConnell ally, met with a small group of donors in New York City. She described the primary as extremely close for candidates Blankenship, Jenkins, and Morrisey. And, according to one person who was present, she said that Republicans were concerned that should Blankenship win the nomination, Manchin would walk over him in a general election.

In an interview on Tuesday, Capito said she would not endorse in the primary and said “it’s really hard to predict” who will win. Limited public polling in the race has indicated that all three candidates are in contention and there is no favorite.

“It’s tightening up, by all indications. And I think that’s going to heighten the interest of a lot of Republican and independent voters in the state and I think that’s a good thing,” Capito said.

It’s no surprise GOP senators are searching for a silver lining, because they find themselves in a nearly impossible position in the West Virginia race.

During last year’s Alabama Senate race, the national party — led by the McConnell-aligned Senate Leadership Fund super PAC — spent millions of dollars in an unsuccessful effort to stop former state Supreme Court Justice Roy Moore from winning the nomination. The barrage failed, with Moore portraying himself as a victim of a GOP establishment bent on taking him down — a message that resonated with the state’s conservative voters.

Moore beat former Sen. Luther Strange (R-Ala.), but then lost to Democrat Doug Jones, shaving a seat off the Senate GOP’s already thin majority. Thune admitted that the GOP getting directly involved in West Virginia would “probably” help Blankenship similarly.

“I still think in the end people are discerning enough that they’ll figure this out. Obviously we’ll want to have a good candidate out there that will run a good race in the fall,” he said.

Instead, Republican leaders are taking a more guarded approach. They have refused to take credit for the recently launched, anti-Blankenship ad by the so-called Mountain Families PAC. Yet their fingerprints are all over the generically titled group, which has enlisted a team of veteran Republican strategists who have worked closely with Senate Leadership Fund in the past. The super PAC’s ad-making firm, McCarthy Hennings Whalen, has also done work for McConnell.

Trump appeared at an event in West Virginia this month and sat next to Jenkins and Morrisey. Blankenship did not attend. Senate Republicans privately said they hope Trump will attack Blankenship if upcoming polls show he could win.

Blankenship, meanwhile, is taking a page out of the Moore playbook. In a scathing statement Monday, he derided McConnell as the “Swamp captain” of D.C., said he was wrongly accused in the mine explosion and compared McConnell to a Russian interloper.

“West Virginians are aware that McConnell cannot vote in their election,” Blankenship said. “They want him to mind his own business and do his job.”