I tried to solder the whole thing together finally and ran into some troubles, basically it didn't work. I don't know what was different from the first time I clipped it all together but I got no sound out of the mic. So I ripped out the funny circuit and tried to splice it all together. That didn't work either on the first try because the phone cord has funny copper-wrapped nylon wires and they don't solder well. In the end I did the unthinkable: I twisted the wires together and shrink-tubed them. This worked better than anything I tried before and the handset is what I will call complete. Actually I'm going to want to get some bigger shrink tube (preferably black) and cover up some ugly spots before I call it done, but I probably won't write about that.
In summary, this was a fun project and I've got a really nice phone handset on my computer now. It cost me very little and wasn't terribly hard. I'm happy with the results.