Miniraise, assuming you have the best hand, seems like he is testing the waters this time, trying to play conservative with his hand and try to get a read to see if you really do have "KK" or "AA." Now, there is a small chance he is trying to trap by getting you to think this, but with KK on this board, you're gonna lose if he has AA or a set. So, away we go...

Best option is to push now and hope that it looks like you are trying to really represent the Kings again; he probably calls with AQ. I think this is a fine option, better than a probe raise, which may be a little too conspicuous.

Or, you can just call, and assuming the turn is not a scare card, bet 2/3 the pot. Maybe he will think you have JJ here and push.

Your only mega-problem card is an A. I'm not sure what you would do here. Probably bet the pot, if he raises all in now, I'm probably making a crying fold.

Argun