It turns out that the professionals use a larger snake, and cleaned out the pipes quickly with his larger tool.
As I type this, I realize that there are many, many jokes that can be made out of the plumber using his larger snake to clean out my wife’s pipes while I was at work. At any rate, the bill was $165 and I can again use my toilet. The best part? My in laws and wife spent the rest of the day cleaning the house, so when I got home everything was already cleaned up.
I don’t think that this was related to the mystery that is the pipes not working when the wind blows out of the north. I think that this was a different problem.