Then I noticed the instrument panel was displaying a message like "Low Beam Failure" or something like that. Yes, the instrument panel displays messages. This is a BMW, you know. But the low beams were working just fine. Hmmm...
So. I made a bold move: I removed the wire running from the headlight switch to the headlight, and removed the kludgey jumper in the fusebox and replaced it with the proper two fuses. Now to try the headlights...
And they don't work. No surprise there. The high beams work, but the low beams almost don't work entirely not at all. Almost. Any significant tap, bump, thump or vibration of the car makes the lights briefly flicker. Ah hah! A bad connection! This is going to be easy.
I know what you're thinking. At this point, you're expecting a long tale of trials and tribulations, whilst I search high and low for a bad connection that simply won't be found, and finally give up and put the jumper and kludgey lead back in.
Sorry to disappoint. Actually, it took me about a minute of judicious tapping around the dash, fusebox, and headlights to determine it was the Light Control Module (aka, the LKM) in the fusebox. Even the lightest tap on the LKM made the lights flicker. So I removed it, popped its lid off, and examined the guts.
It's a printed circuit board with a row of relays soldered to it. Here's a tip: If you have a printed circuit board with a row of relays soldered to it, eventually the solder on the relays will fail. It will fail. Always. That's because the combination of the weight of the relays, the heat produced by the relay solenoid, the vibration of the car, and the clicking of the relay contacts will inevitably stress the thin solder until it breaks. Always.
The fix is simple. Resolder the relays, like this:
Here's a close-up of the relevant connections. You can't see the breaks in the solder -- they're too thin to be visible to the naked eye.
And that fixed it. Boy was I tickled! I figured I'd found a clever but previously unknown tip to fix E34 headlights, and the entire used BMW community would celebrate my name and thank me by taking up a collection to buy me an E34 M5 and and an E24 M6, but then I found out that this is already a well known solution, documented at http://www.bmwe34.net/E34main/Maintenance/Electrical/LKM.htm. Oh well.
Then I pulled all the dysfunctional alarms out from behind the driver's side footwell and tossed them in this box:
In other electrical trials and tribulations, I found out why the electric mirror adjustment doesn't work. Burned out fuse? No. Burned out relay? No. Missing relay? Yes. Sheesh. This usually happens when some more important relay -- like the horn relay -- fails and gets replaced with a less important relay, like the electric mirror relay. Tsk. Tsk.