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Um, I might be pissy since I got my ass handed to me this AM. However...

Your line plays like you're convinced you're ahead, and I don't think you should be so convinced...

You should have raised the flop to try to keep control of the hand there. If he re-raises you, you should suspect you're behind. You don't know where you are after that point.

Turn card is worst possible for us. When the second K comes and villain leads into you, he's telling you he thinks you don't have a K (and he was right) and that he can beat AK. However, since you didn't raise him on the flop, you don't really know where to gauge his hand. You are now stuck really stuck. Given how the hand played out, you should have raised him and folded to a re-raise. You probably could have shown down for free and/or reacted to the river card from there.

To me, your river raise is spewing. There are plenty of hands you lose to, and not many you beat, and he clearly wasn't going to fold to your raise, so there was no fold equity in the raise.

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He can beat AK? So he has AA? Raise the flop, get three bet by a worse ace and cap. I would just call his river raise.