To make room for Mr. Ayers, Mr. Trump, who famously avoids one-on-one interpersonal conflict, had been trying for a while to pull the trigger on firing Mr. Kelly. Famous for the “You’re fired!” catchphrase and also for hating confrontation, Mr. Trump had looked for others to do the work for him last week — even attempting to arrange for Mr. Ayers to fire Mr. Kelly — according to three people familiar with the events.

Finally, Mr. Trump persuaded Mr. Pence and Mr. Ayers to join him in hashing things out with Mr. Kelly in the presidential residence on Friday night. But instead of sticking to the plan to let Mr. Kelly leave with dignity, which Mr. Ayers and others in the White House had urged the president to do, Mr. Trump decided to announce it himself on Saturday.

Finding a replacement for Mr. Kelly is in many ways just the latest staffing snare in a White House that has struggled to fill even low-ranking jobs. Staff turnover for senior aides, according to the Brookings Institution, sits at 62 percent.

“After two full years, Obama was at 24 percent turnover of his senior staffers,” Kathryn Dunn Tenpas of Brookings said in an interview. “George W. Bush was at 33 percent. In a way, instability breeds instability.”

But the chief of staff job is often described by those who have had it as the worst in Washington, even when times are good.

Leon E. Panetta, who had the job under President Bill Clinton, said that the Clintons squired him to Camp David to work on him to leave his job as the director of the Office of Management and Budget. Mr. Panetta recalled Mr. Clinton’s clincher:

“‘You know Leon,’” Mr. Panetta said Mr. Clinton told him, “‘you could be the greatest O.M.B. director in the history of the country, but if the White House is falling apart, no one will remember you.’”