Donald Trump summoned senior aides to his 26th floor office at Trump Tower around 2009 for an unpleasant task: persuade Michael Cohen—a top lawyer at his company—to resign, according to one of the aides.

Mr. Trump’s frustration had reached a breaking point, the aide said. Mr. Cohen wasn’t getting things done and Mr. Trump didn’t believe he was a good fit at the Trump Organization, even though he liked him and didn’t want to fire him. “What’s he doing here?” Mr. Trump demanded.

The...