I got out of the prison, met up with one of my former captives, and made my way to the place where the bad guy lowers the drawbridge. There's no way to follow him up there, so I wait until he jumps down, kill him, and grab the key.

Yet when I try to exit the door, it keeps telling me I need a key to open it. I thought the key was the one I just picked up. Is key use automatic or do you have to select it somehow and then use it?

I'll give that a shot. Maybe I screwed up the key configuration. It's a small room with only one way out (the door).