Got it off using a bit of elbow grease and some heat.
First, to get the majority off we put ice cubes around the plastic on the coils and used a piece of hard wood to scrape it off. Metal can apparently damage the coils beyond repair. After the majority was off, we ran the washer a couple times melting the plastic then using a wet cloth with acetone and water to rub off the remaining goo.
It still smells horrible, though, so we are running it a couple more times with dishwasher cleaner and vinegar to clear that out.
Moral of the story: Secure your shit in a dishwasher.