Well, this project is becoming a bit more work than first hoped. I reassembled the carb and reinstalled it on the engine. Then I gave it a few pulls and this...
So, I got some rope and fixed the recoil. That was also a pain because the recoil spring came loose and went flying across the room. Youtube to the rescue to show how to easily wind that thing back up.
Anyway, I reinstalled the recoil and gave it a few tugs. No results. A little starting fluid started it right up, and then it died again... So, the carb is back off and soaking again. If this next attempt doesn't work I'll probably order a new carb. It just really seems like a waste for not being able to clean a hole out.