How do you play this?

[PlayerA]

I'm in the $5 re-buy satellite to the 500+30 on Stars. I have 36K with 37 players left. This puts me in about 27th or so. Most hands not seeing flop. The blinds are 1K/2K. I am in EMP with K [img]/images/graemlins/spade.gif[/img] J [img]/images/graemlins/spade.gif[/img]. I make it 6K to go. Player in LP who has me covered flat calls. BB calls.

Flops comes 6 [img]/images/graemlins/club.gif[/img] 9 [img]/images/graemlins/diamond.gif[/img] 9 [img]/images/graemlins/club.gif[/img]. BB checks. I check. The LP player makes a weak 3K bet into the over 20K pot. No real read except that it looks weak, smells weak. What's your move?

I check-raise to about 12K. LP thinks and calls. BB folds.

Next card is 10 [img]/images/graemlins/club.gif[/img]. Giving me a gut shot provided I'm not drawing dead. I move in and he calls and shows A [img]/images/graemlins/diamond.gif[/img] K [img]/images/graemlins/club.gif[/img].

This hand was winnable for me without a doubt. However, I think went it about all wrong. I'm just curious how others play this post flop (I don't care if you think KJs is trash and blah blah blah).