But three things put off Ms. Sebelius’s departure: Mr. Obama’s fear that letting people go in the middle of a crisis would delay fixing the website; his belief that ceremonial firings are public concessions to his enemies; and the admiration and personal loyalty that Mr. Obama still felt for Ms. Sebelius and her advocacy for his chief domestic legacy.

Over the next four months, Ms. Sebelius engaged in a kind of slow-motion resignation, largely staying out of the national limelight but crisscrossing the country in a furious effort to enroll people in health insurance and taking comfort from strangers who recognized — and offered thanks — for her efforts.

As the website improved and enrollment numbers neared the administration’s goal of seven million people, she began plotting her exit.

“My balance has always been, When? How do you make that decision? I have had conversations off and on with the president about that very thing,” Ms. Sebelius said in an interview on Thursday. “It became more definite in early March, when I felt confident that we were well on our way to a robust enrollment.”