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Great first post.

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I would reraise him all-in.

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He pushed.

Yes but if you reraise with all your chips it looks more assertive and you still get your chips back.

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He's trying to steal.

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How do can you be so sure?

Gut feeling. And I guess trying to steal is the wrong word. He's trying to steal, but he still has a decent hand. He's a got a medium to top hand, and he's looking to win without a fight. He's hoping you don't call and he takes the pot, but he's also thinking he has a pretty good chance of winning (because the Ks were slowplayed) with his hand. He may have QQ or AJ.

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Also I would have bet 5xBB after the flop and found out what he had then.

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Say he calls. What does he have?

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That means he had something decent right after the flop. If he called then it probably means he had the AJ. If he had the QQ he would have raised thinking he had top pair. If he only called, then you would be in luck, because you'd see the K on the turn and be ready to fight some more knowing you probably had the best hand. Instead of just guessing on what he has at the river.