I tried several software fixes and then dug into the hardware. I even tried an external USB keyboard to no avail. After buying a replacement laptop I decided it would be a good idea to try removing the keys completely, including the board. That was a bad idea.
It didn't solve the problem and it created a new one. To make matters worse that laptop is where my Adobe software lives and I've long since lost the discs and things needed to migrate it to my new laptop.
I finally decided that the real solution was going to be installing a new keyboard and then doing a system restore.
Step 1: Finding the right part
From the looks of things, if you have a Toshiba laptop with a broken keyboard it's a mildly intimidating but simple thing to replace.