David, thank you so much for the tape suggestion! A little Scotch tape just ended 15 minutes of eyes-widening frustration. :)
I ran into trouble when re-installing the keyboard cable. I got everything put back together, plugged it in, and pressed Power... and nothing happened. I took everything apart again, reseated the heat sink and everything, put it back together... and the Power button didn't respond.
The MagSafe light came on, and the battery charge lights came on. But it wouldn't power up.
Finally I figured out that the keyboard cable is really tricky to get in correctly. If you can slide it out with the retaining clip locked (applying very little pressure), then the cable isn't actually in correctly. I had to really fiddle with it, and maybe slide it in one side first, to see that it actually needed to slide in another 1/8th inch or so beyond what felt like "in" the first few times I did it.
Once I'd secured that cable, my MacBook turned on again without a problem. So far, after just replacing the Thermal Paste (I didn't actually install a new heat sink), the computer is running 20°F cooler than it has in years. Thanks!