[AaronBrown]

You can make it a little easier on yourself. You have seen six cards, so there are 46 unknown ones. They can be arranged 46*45*44/2 = 45,550 ways.

But for most purposes, only the ranks matter, not the suits. There are only 1,183 different rank combinations. Only if the KJ hand wins do you have to ask if all three of the new cards (the other player's hand plus the river card) are clubs or hearts. It's easy to do that as a side calculation.

Within the ranks, you can simplify further. There are two main possibilities for KJ, it pairs one of those cards or it doesn't. There are four possibilities for the unknown hand: no pair, one pair, two pair and three of a kind. Only in the no pair/no pair and one pair/one pair cases do you have to analyze further to see who wins.

Finally, you have to figure out the various straight possibilities for the other player's hand. There aren't many of these.

You could do this with pencil and paper in half an hour or so, by enumerating cases as above. Or you could spend about the same amount of time writing a computer program to do it in a few milliseconds.