I like this hand, but I think it's hard because it doesn't matter. I'll explain.
I think on the flop you can call, as you did, and you can also raise 1/2 pot, and I don't think it matters which play you make and both are fine and you could probably mix them. You need to be raising some boards like this with hands like yours. I think, with diamonds, I lean more towards call. If I had naked K
I lean more towards raise, but both plays are valid.
On the turn - raising makes less sense and is less congruent with the total action, but you simply cannot fold so you have one action remaining.
River. You are likely indifferent between your decisions here. Flophero likes a mix of calling and folding, leaning towards calling. Your opponent likely has AA, it's just his most likely holding from a raise UTG, especially when you have KK yourself, but he can also have a 4 or a 3 (Ace + two broadway single suited is a raise UTG, so any 3 or 4 is definitely possible). He can also have the occasional 56 and diamonds are of course also quite likely. You can run the percentages a bit and see this.
Now, up until the river I think his hand is very consistent with AA but you have so much equity it doesn't matter - you have to get to the river. But now, is AA as likely? He should be check/calling a lot on this river IMO, so I would weight him more towards bluff / 4 and I think he has more bluff combos then 4 or 3 or 99, but i also think people bluff less often than they should.
This is a very long way of saying I don't think it matters much what you do here. I probably lean towards call, as you did, but I think folding is fine as well. I think it's a spot where villain becomes naturally balanced due to the run out and typical player tendencies and I think you have a threshold calling hand and that's just the way this hand played out.