When he finally went to bed early in the morning on Oct. 2, Sean Hannity had a good sense, as he typically does, of how he would structure that night’s Fox News Channel broadcast. He’d lead with Puerto Rico, and a defense of the Trump administration’s hurricane relief efforts, before moving on to the N.F.L. players who continued to kneel during the national anthem before games. But by the time he woke up, a few hours later — Hannity rarely sleeps more than four hours a night, a trait he shares with his friend President Trump — the screen of...