Kings vs. Flush Draw

Preflop, I raise 5x (there was an early position limper). The flop brings a heart draw and I fire off just over a 1/2 pot sized bet which the button calls. I put him on AhTh, KhQh because I think he reraises preflop with AhKh, maybe even AhQh from the button.

How would you have played this differently? If I slow down, I give him the pot odds to draw out on me. Should I have made a full pot sized bet off the flop? Again at the turn?

PokerStars No-Limit Hold'em, $1.00 BB (6 handed) - Poker-Stars Converter Tool from FlopTurnRiver.com

Button ($97.25)
SB ($99)
BB ($47.70)
UTG ($49.75)
Hero (MP) ($101.45)
CO ($84.90)

Preflop: Hero is MP with K, K
UTG calls $1, Hero raises to $5, 1 fold, Button calls $5, 3 folds

Flop: ($12.50) 7, J, 3 (2 players)
Hero bets $7, Button calls $7

Turn: ($26.50) 2 (2 players)
Hero bets $15, Button calls $15

River: ($56.50) J (2 players)
Hero bets $15, Button raises to $30, Hero calls $15

Total pot: $116.50 | Rake: $3

Results in white below:
Button had A, 6 (one pair, Jacks).
Hero mucked K, K (two pair, Kings and Jacks).
Outcome: Button won $113.50

PS: Hand Converter seems wonky with this hand. See results.

You'll get better responses without including (and in this case advertising) the results.

My feelings on this: pre-flop is fine, bet more on the flop, bet more on the turn. As played through the turn: check/fold the river unless he bets small - the top card paired & the backdoor flush got there too, so you're not beating much except a bluff. As played through his river min-raise: fold there as well, for the same reasons. He's not min-raising you because he has a pair of sevens and thinks you'll fold your trip jacks or overpair.

Thanks. Hadn't realized the card images were visible in the white results. sort of ruins the surprise!
So make a PSB on the flop. If he calls it he's either hit the set or looking to suckout or a fish. As played, looks like I'm the fish

Yeah - he can have a flush draw or a couple of gutshots on the flop, so I'd make at least a $10 bet there and make it pricy for him to continue. On the turn 2/3 or 3/4 of the pot is probably right, especially with a larger pot going (and now only one card to come - and worry about). Take him to valuetown.