what is the benefit of waiting until almost to end a speech to ask Congress to declare a state of war

The final punch comes at the end.

Explanation:

Let's say you want to convince Congress that war is necessary and inevitable.

Now although history is full of unnecessary wars, in general people and Congress alike try to avoid it. Therefore your argumentation has to be ground-solid and convincing.

If you start by saying that war is necessary you immediately turn possible voters in favor of you, against you. So the best way is to prepare the Congress by a well-balanced speech in which you don't even mention the word. Not until the end.