Where Sam doesn't play anymore ...Rick's Cafe Americain

Problem is, the guy in second position (who limped in, remember) calls all-in.  I take back about $3,000, and the cards are turned:

ME: K♠ K♦

Number 2: A♣ J♠ (!)

Number 3: Q♣ Q♥

I’m feeling pretty good since I was expecting 2 to have AA.  The flop comes, and a J hits, but so what?  I’m still a huge favorite to win (huger, in fact, now that there are only two cards to come).

So what is the A♠ on the turn, is what. 

Now, at this point I’m resigned to losing, but there are three spades on the board, so with the K♠ in my hand and the J♠ in Number 2’s hand, I’ve still got 9 outs.  Right?

Obviously, I didn’t improve and got knocked out on the next hand by 88 when I went all-in with A5s against A8o.

Now, Two Dimes says that my chances of winning that second hand were less than 30%, whereas my chances of winning the KK hand were 58.17%.  I’d be inclined to believe them, except they also claim that I had 10 outs going into the river:

pokenum -h kd ks - ac js - qc qh —2s jc 9s as
Holdem Hi: 42 enumerated boards containing As 9s 2s Jc
cards win %win lose %lose tie %tie EV
Ks Kd 10 23.81 32 76.19 0 0.00 0.238
Js Ac 31 73.81 11 26.19 0 0.00 0.738
Qc Qh 1 2.38 41 97.62 0 0.00 0.024

I thought this might be because it was missing the J♠ in my opponent’s hand, except it correctly limits Queenie’s outs to one, since Q♠ would give me the winning flush.  Any thoughts?