Are you putting him down or is he dying? You changed the title...
My dad said he could put her down but I've had a bad experience putting another pet down. It went wrong and it just put the pet in more pain from where it didn't kill them. So there isn't really much to do but wait for her to die. It's really disturbing because half an hour ago she was doing ok. Now she's just paralyzed on her back breathing heavily and trying to move.
Seriously not to sound like a richard or anything, but just kill it already, you're just putting it through more pain now, get a small gun or something, hit it right in it's head, the thing won't live through it.
That's too gory.This isn't a loving doge.
There really isn't any efficient way to put down a hamster that I know of. I've had to do away with chickens, and that is hard to do. A hamster would be worse, and I'd fear for doing something terrible.