I got a rebuilt kit from the auto parts store for $25. It had an exploded parts diagram and the parts were numbered in the order that it should be dis-assembled in. So I stepped through them one at a time and complete took it apart. I kept the small parts grouped together and stored in ziplock bags labeled by the part numbers from the diagram.
As with the painting I tried to avoid solvent based cleaners as much as possible. I found that Goo-Gone in a spray bottle did a good job of getting off most of the grease and grime.
I used boiling water with Simple Green degreaser to rinse the parts.
Once I got all the dirt and grease off that I could, I cleaned them in carb cleaner. This carb cleaner comes in a 1 gallon bucket. It also has a coating of oil on top to keep the carb cleaner from evaporating. After about 5 hours of cleaning parts, it looked almost as good as new:
Then I just followed the parts diagram in reverse order to put everything together. The rebuild kit replaced all non-metal parts as well as a few metal ones.
I'll still need to do quite a bit of tweaking and adjusting before I can fire up the engine again, but I think the majority of work is done.