Just an update on this problem. I brought my car back to the windshield shop, and I talked them into letting me hook up my rain sensor to a new windshield (with my homemade rain sensor extension cord). The rain sensor worked just like it should on the new window, so the shop replaced the windshield for me again, this time free of charge.
I think that I might have been able to get the old windshield to work if I could have gotten the sensor re-initialized, but that would have cost me $125 at the BMW stealership.
Anyway, it's all working perfectly now. Thanks for the help.
2003 325xi BlackČ