I did try TopNotch's method. No dice. I tried the freezer method. No dice.
Soldering is not my strong suit, but I'm going to try swapping the drive electronics. I really only want the stuff that was in my downloads folder, but it isn't the end of the world if I don't get that stuff.
Meanwhile, the laptop is reloaded with the new drive. OS, drivers, FTP application, graphics applications, Office applications, browsers, backup software. The restore/sync is almost complete. Most of my settings are in place. PDF applications are installed. I had to change some fonts. I missed a few browser tabs.