White House aides insist that the Kelly memo is not just a bid for better headlines — it would dramatically improve a clearance process that had gotten out of hand, they said. But what Mr. Kelly has not publicly proposed is investigating precisely how the unsettling episode occurred.

There would be much to investigate. A few senior officials in the West Wing had access to details of the allegations of physical and emotional abuse made by Mr. Porter’s ex-wives. A greater number of people had a sense that something was amiss, but chose to avert their gaze instead of asking questions.

The first hours after the issue was brought to light by an article in The Daily Mail were a confused blur, with officials like Mr. Kelly at first endorsing Mr. Porter’s resignation, but then agreeing to issue statements in support of him. That disarray belied the belief of some in the White House that someone — Mr. Kelly — was finally in charge after six months of drama and infighting in 2017. In reality, these officials now concede, no one truly was.

“They haven’t figured out how the place operates, and apparently they don’t want to learn,” said John Dean, a White House counsel under President Richard M. Nixon. The Porter situation, he added, “is a manifestation of what happens when you have chaos.”

Mr. Kelly, a four-star Marine general, had been billed as uniquely qualified to bring order to Mr. Trump’s world when he took his post in July. Now he faces a morale crisis in the West Wing, where aides describe a sense of betrayal by a chief of staff they no longer trust after his claims that he had not fully known about Mr. Porter’s problems and had acted within minutes once he learned of them.