With the rocky rollout of his health care law in a post-deadline lull, President Obama will move this week to shake up his health care team, accepting the resignation of Health and Human Services Secretary Kathleen Sebelius and tapping his budget director as a troubleshooting replacement, the White House confirmed Thursday.

Mrs. Sebelius, a former Kansas governor who has served as HHS secretary for Mr. Obama’s entire tenure, was the president’s point person as his signature law stumbled out of the blocks in October.

Despite calls for her to be fired, Mrs. Sebelius stayed on the job to oversee a recovery of HealthCare.gov that resulted in at least 7.5 million enrollments for health care plans.

Mrs. Sebelius, however, is seen as damaged by too many congressional Democrats who are warily eyeing November’s elections, in which Obamacare is expected to play a major role.

Some Republicans could barely contain their glee.

“Are you going to be drinking out of a red solo cup or crystal stemware tonight as you celebrate @Sebelius resigning?” Rep. Marsha Blackburn, Tennessee Republican, wrote in a Twitter message. She was making a reference to a sharp exchange she had with Mrs. Sebelius at one of the congressional grillings the secretary faced after the botched rollout.

Senate Minority Leader Mitch McConnell also alluded to a previous Obamacare-related quote, in his case a famous comment from former Sen. Max Baucus, Montana Democrat.

“It’s fitting that nearly one year after the primary legislative architect of Obamacare predicted it would be a train wreck that the government official most responsible for overseeing it reportedly is resigning,” the Kentucky Republican said before going on to call the resignation “cold comfort to the millions of Americans who were deceived about what [Obamacare] would mean for them and their families.”

The White House said Mr. Obama on Friday will tap Sylvia Mathews Burwell, director of the Office of Management and Budget, to replace Mrs. Sebelius.

Ms. Burwell took the budget post a year ago after winning confirmation on a 96-0 vote.

She will have her work cut out for her at HHS as the department prepares for the second round of Obamacare enrollments later this year and as it tries to limit the disruptions, price increases and cancellations that Republicans say are bound to occur.

Already, insurers are reporting that a statistically significant percentage of people who selected plans on the Obamacare marketplace have failed to pay up. On Thursday, an executive at PacificSource Health Plans told The Washington Times that an average of 9 percent of would-be enrollees it obtained from the exchanges in Oregon, Idaho and Montana have been canceled because of nonpayment.

Meanwhile, the federal exchange system known as HealthCare.gov is transitioning to a new vendor, Accenture, and several state exchanges are switching Web platforms or battling their contractors over payment, citing poor performance.

Sen. Joe Manchin III, a West Virginia Democrat who has been among the biggest Obamacare critics within the president’s party, said Ms. Burwell is the right person to oversee the fixes needed to keep the law afloat.

“Sylvia’s experience in both the public and private sector, matched with the bipartisan relationships she has built over the years, shows that she is a public servant ready to take on this country’s challenges,” he said.

Dan Mendelson, CEO of the Avalere Health consultancy in Washington who worked with Ms. Burwell during the Clinton administration, said she is the right pick for the job and “confirmable.”

Senate Democrats should have an easy time approving the president’s choice, now that it has employed the “nuclear option” and can approve nominees on a majority vote without having to overcome the 60-vote threshold to avoid Republican filibusters.

Mr. Obama stuck with Mrs. Sebelius despite repeated calls for her resignation late last year.

But during a Rose Garden announcement last week to tout more than 7 million Obamacare enrollments, the president didn’t mention Mrs. Sebelius, even though she was seated in the audience in front of Mr. Obama.

She submitted her resignation just days after the March 31 sign-up deadline.

“Few Cabinet secretaries go the full eight years, but it’s good that Secretary Sebelius is able to resign at a time when implementation is going well rather than during the dark days of last fall when success seemed far from certain,” said Timothy Jost, a health care law analyst and a professor at Washington and Lee University School of Law. “Ms. Burwell seems well-positioned to move implementation forward.”

House Minority Leader Nancy Pelosi, California Democrat, said the secretary’s legacy will be the more than 7 million Americans who signed up for health care plans and the more than 3 million who now qualify for Medicaid under the Obamacare expansion.

“From day one, Secretary Kathleen Sebelius has remained laser-focused on a single purpose: to make health care a right, not a privilege, for all Americans. Her leadership has been forceful, effective, and essential,” Mrs. Pelosi said.

Mrs. Sebelius‘ resignation also will be seen in the political context, and the Republican National Committee was among the first in Washington to react to the news of Mrs. Sebelius’s departure.

“Secretary Sebelius oversaw a disastrous rollout of ObamaCare, but anyone can see that there are more problems on the way,” said RNC Chairman Reince Priebus. “No matter who is in charge of HHS, ObamaCare will continue to be a disaster and will continue to hurt hardworking Americans.”

Obamacare’s ill fortunes marked a low point in Mrs. Sebelius‘ career, attracting fury from Republican lawmakers and spoiling her relationship with a fellow Kansan, Republican Sen. Pat Roberts, along the way.

Mr. Roberts’ challenger in the GOP primary this year, Milton Wolf, issued a statement Thursday attacking Mr. Roberts for having supported Mrs. Sebelius.

Mrs. Sebelius‘ resignation was announced hours after she appeared before the Senate Finance Committee to testify about her department’s 2015 budget request. She offered no hint of her big move.

Instead, she touted Obamacare enrollments and progress in the Medicare Advantage program.

At one point, Sen. Johnny Isakson, Georgia Republican, said he looked forward to working with the administration in “the years ahead” to put health care entitlements and Social Security on a stable path.

“Well, I would very much look forward to that opportunity,” she replied.

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