Enemy A.I. won't die

[Alneyan]

Are you playing a Simultaneous game (where your orders are executed at the end of the turn) or a Tactical game (where your orders are immediately executed)? If it is the former, the battle should occur when the turn is proceeded, and you should have the battle report when you play your next turn.

If it is a Tactical game, a window should prompt you whether you want to handle this battle yourself (that is a tactical battle, where you command your ships on your own) or leave leadership to the AI (strategical battle).

I apologize for asking this question, but are you sure the ships/planets you wish to capture/destroy belong to the enemy? I made this mistake once, and desperately tried to destroy my *own* ships. No wonder it didn't work for me then. As you have been using Intelligence, you might have taken possession of former enemy ships, even if these keep their old look. Hope it helps!