AA - did I play it right?

In a 10/20 bricks-and-mortar game, I had AA under the gun.
The table was aggressive, so I opted to limp, hoping to limp-raise pre-flop.
The field didn't disappoint, as a weak player bumped it; the big blind called, and I three-bet the pot.
They both called.
The flop was K, K, 3.
The big blind checked, I bet for information, and they both called.
The turn was 5, the big blind checked.
Thinking the pre-flop opener was likely to hold AK, I checked. To my surprise, she checked behind me.
The river was a King, and the big blind checked.
I bet, the late position player called, and the big blind raised. Thinking I could beat a lot of holdings, I called, as did the late position player (she had 77).
The big blind turned over KQ.
Four of a kind beats kings full of aces. Kings full of aces doesn't even qualify for most bad beat jackpots.
I could have bet the turn; I could have checked the river; I could have folded after being check-raised on the river; I could have open-raised with my aces. Poker is filled with 'could haves.' Rather than shut out the field and win the blinds, I tried to trap my opponents; it didn't work out. That doesn't make it a bad play, it makes it a bad result. The goal is to win the most money, not to take down the blinds or win the most pots. And, I love the chance to play in a limit game for three bets when I hold Aces and my opponents hold 77 and KQo!