So, his approval ratings are sinking.

What makes you think that he, or anyone else in his administration, cares?

And I agree with that. Sometimes a head of state or any leader has to make unpopular decisions because those decisions are right (or believed so).

Even in his last days, when Germany was in ruins, people were dying without necessity, and he was living in a bunker in Berlin, Hitler continued to do what he felt was right.

It was only when the Russians were nearly literally at the door that he realized the truth and killed himself.

Isolate the leader. Don't let him or her have any contact with anyone who doesn't agree with them. Keep them convinced that the chain of command is God Almighty --> The Leader, who has been personally chosen by God.

Feeding megalomania has its drawbacks, though. There's Hitler, and several Caesars, and Marc Antony, and Oedipus, and Dugout Doug MacArthur, and I'm sure that you can add to the list.