How do I know what examples to use in a depth essay?

Because it is a depth paper you should aim to use or at the very least mention a few examples in every paragraph. But try not to panic and fill it will irrelevant examples. Its better to cover the whole time period with specific examples to show the examiner that you do understand the question. You can usually start by "book ending" the question with key dates or examples. So if you were asked, "To what extent did Henry II restore royal authority in England between 1154 and 1166?", you should start by coming up with some key dates in and around those two years given. For instance, it was in 1154 that Henry became king, but his position was legitimised in 1153 with the Treaty of Winchester. Then in 1166 there are multiple dates that you could mention including the Cartae Baronum and assize of Clarendon, though if you think it is relevant to the question you could also briefly mention any statistics or laws from 1167/8 or the 1170 Inquest of Sheriffs.
While that's a good place to start and will help you frame your answer, you'll benefit most from having already learned key dates grouped by theme. If that's something you want to try we can work on some mini timelines that focus on recurring themes in the subject so a timelines specifically for: Royal Authority; the Church in England; Henry's relationship with his sons and so on... I'm afraid there's no easy way to get around learning facts and examples, but we can at least break down how you think about them so when a question comes up you know exactly what to write about.