I managed to find and order the parts I needed. Two STKs and a pack of thermal paste, total including shipping was about $50. These are the steps I took to fix the TV.
Prior to this I've actually never used a soldering iron, I was fully prepared to have to buy a new TV by the time I was done with this, so I was both surprised and pleased with the results.
A big thanks to the creator of this article:
And the people who help from this one:

This is a 51 inch Sony rear projection TV
Model: KP-51WS500
Manufactured on September/2002

To work on the board I had to unplug several wires and remove some screws. To make life easier, I put a label on every disconnected wire and the spot it came from. The pictures were also a big help. I started at 1 and worked my way up, but I am going to re-order the pictures to be what I would consider an easier order to the process.

The most important step of all, UNPLUG YOUR TV!
I actually unplugged mine before going out of town on a business trip, so it was without juice for almost a week before I started working on it. On one hand, that doesn't mean you're 100% safe, there are still capacitors that can hold a charge for a while. On the other hand, this isn't a CRT TV, so not as dangerous I guess... still, be careful!

Please note I do not promise or guarantee ANYTHING AT ALL.
Doing these steps may not fix your TV.
Doing these steps may void any sort of warranty you have for your hardware.
Doing these steps may injure or kill you.
By following these steps you take full responsibility for the results.

I now realize I didn't take any pictures of the "remove the back panel" step... but I think if you're having trouble with that part, maybe you should have a friend help :)

We are unplugging the black wire, this is right next to the lamps. And while we're here, take out the piece of particle board at the bottom of the picture.

This will be on the left side

Along the front.

I didn't do this till near the end... I suggest doing it here though, take out the board on the left side. After this screw is removed it will slide towards you.

And remove the wire that feeds it power

Lots of screws to take out...

Left of the big heat sinks up front.

Behind the big heat sink.

To your left.

And to the right.

Yes, there's a screw behind that heat sink. This is one of the reasons why that board on the left has to come out.

Now these things, no idea what they're called, but there are 4 of them on the right side...

Simply lift up on them, may need to rock your fingers to pry it up.

Along the front of the "D" board (that big one we just took all the screws out of) are these little clips you need to squeeze with your fingers. I started out on the left...

And then propped up the corner with a screw driver to stop it from snapping under the clip again. Then I undid the other clips (I think it was 3 total, all along the front)

I decided to disconnect only the wires that were preventing me from standing the board up inside the TV, rather than do all of them and totally remove the board.

These were at the back on the left.

On the right

And here is the board, finally able to stand up inside the TV. I have circled the solder points of the STKs in blue.

Close up of the left.

Close up of the right. Don't ask me about the red, it was there when I got here.

When I removed the solder, I never did 2 points in a row, I'd constantly move from one STK to the other, to help prevent overheating. I made sure each leg could wiggle freely before considering the job done, here are my results...


Each STK has 2 screws holding it to the heat sink, remove them, and pull out the STK. Oh, and this part is easier if you set the board back down in the TV.

Here are my new STKs. I may have put a bit too much thermal paste on these... better than not enough though, right?

Slide the little legs through the holes, and screw them in place. It should be noted that the original STKs were not screwed very tightly against the heat sink, so I made sure to do the same with my new ones. I tightened till some of the paste oozed out, but did not screw them all the way in.

And again, while soldering, I made sure to move back and forth between the left and right, to help prevent overheating.
Here's the left

And the right

After I was done soldering, I went backwards through my steps, making sure everything was plugged back in where it belonged (didn't do the screws though, just-in-case I needed to pull it all apart again). Turned on the TV, did a flash focus, and everything looked pretty darn good! I'm very happy with the results.

There is one minor issue though; the blue and red are slightly out of focus. I can hardly tell at all when watching normal TV, but when I throw up a test image with red/blue/green lines, it shows. I'm going to post about it, hopefully it's something that can be easily adjusted, but if not, I'm still happy with my TV, fixed for $50!

Link to my red/blue being out of focus