I found this forum by doing a search --- I'm having the same problem. In fact, I have the same 2004 325i, and I'm in Chicago too LOL. I initially thought this was due to the salt (but I didn't get the same issue the first year I was here in Chicago)... I found another forum that suggested the following:
You have to replace your mirror. Use a trim removal tool or you can use a screwdriver but make sure you use a rag around it so you wont scratch the paint. They are easy to change. Make sure you give the dealer the correct year of your car or you will get the wrong one.
MIRROR GLAS HEATED PLANE LEFT 1 07/2004 51167157247 $46.59
ONLY IN CONJUNCTION WITH
ADAPTER LEAD, MIRROR HEATING 1 07/2004 51167836442 $6.58
But I'm hesitant in doing it myself...