So, these mixers are now only about 5 1/2 years old. The photo below shows the general condition I found them in this week; I think that Dante needs to deal with people who would allow this to happen to a KitchenAid:
So, should you ever find yourself in a similar predicament, here's what you'll need: warm water, Dawn dish soap, a gentle rag, a toothbrush, a towel, and a magic eraser.
Wash the unplugged machine as best you can with the water, dish soap, and rag, using the toothbrush for all the little nooks and crannies. Dry with towel. Repeat. Then use the magic eraser over the entire thing to pick up the well worn-in grime you can still feel with your hands. This should do the trick. For a little added shine, spritz with a little bit of glass cleaner. Bask in the glow of your beautifully transformed stand mixer : ).