Since taking the thing apart involved working with complicated connections in extremely tight spaces, I took pictures while I worked to aid in reassembly. While most of the pictures are boring (and weren't helpful, as the dishwasher was easy to reassemble), one came out cooler than I had planned:
Electrical connector inside a dishwasher
We spent about four hours tearing the thing apart (removing the sump and motor assembly, parts of which were held together with Torx screws), fiddling with all the parts, and reassembling it, but it still didn't work. It would fill with water and drain the water, but wouldn't swoosh the water around. Instead it just buzzed.
We thus figured that the motor had burned out or permanently frozen up (despite our efforts to unfreeze it). We proceeded to go online, find the appropriate replacement motor, and were about to order it (literally less than 30 seconds away from clicking "confirm order") when I decided to give our current motor one last chance. I turned the dishwasher on, and within a minute the buzz was replaced with the sound of swooshing water.
So, I have thus discovered the secret to repairing appliances: threaten them with replacement.
Actually, this flawed reasoning is similar to how alternative medicine treatments are sometimes justified. Instead of crediting their recovery to the hard work (founded on years of experience and training) by a qualified medical doctor, patients are all too ready to attribute their newfound health to an alternative treatment they received just before their symptoms subsided. Nope, that surgery and chemotherapy had nothing to do with your cancer going into remission; it was all that carrot juice you drank.
[Update July 2006: Read the next installment of our dishwasher saga here.]