Welcome to the club! Here I am answering your post after pointing you towards the forum in the first place...
If it's just the odometer that died, you probably lost a gear in the odometer. These can be replaced without having to replace the whole instrument cluster, with a little technical skill. There are a few other things that it could be, but more than likely it is just the gear. BMW used a urethane gear on the odometer, combined with a petroleum-based lubricant that causes the gear to degrade over time.
The Bavauto write-up is pretty good. There are also these:
Write-up for replacing the gear:
http://www.odometergears.com/VDO_MotoMeter_Gear_Replacement.htm (also sells the replacement gears)
Another write-up from E30Tech:
http://www.e30tech.com/forum/showthread.php?t=51635The cluster comes out from the front - disconnect the battery, remove the airbag, remove the steering wheel, remove a couple of trim pieces, and there are four screws that hold the cluster in. I know there are folks that have managed to somehow remove the cluster without removing the steering wheel, but I have never been able to do that. There just isn't room.
Key point: do NOT turn on the ignition or reconnect the battery while the steering wheel/air bag are disconnected. The SRS/air bag system will set an alarm, the SRS light will come on, and I haven't found a dealer yet that can turn it back off - the baud rate for the E30 SRS computer is too low for the equipment the dealers have now (at least that is what I have been told).
I know there are shops that do it for exorbitant sums of money, but I wouldn't hesitate to give it a shot myself if you are at all handy. If you're really hesitant about doing it, we can talk - I can probably fit some time in over the next few weeks to do the repair for you, but we'll have to figure out a good time do that, particularly if you don't want to pull the cluster out yourself.