Before I talk about the river I think you need to be getting it in on this turn. He's got half his stack in at this point so he's not folding anything but total air which is going to give up when you call his raise anyways.

If he is a competent aggressive player i don't see many kings in his range that take this line. They will either ck-r the flop or raise your limp pre. So you are looking at a 2 if his hand is ahead of yours most of the time. If you think he can fold a 2 given on his play so far than jam, if not check. You are never getting called by worse here imo.