If you rewire your heated mirrors to use the defroster switch you may lose the temperature/gear selector display. Here's a fix I made for my 97 A4. This may or may not work for you and you are performing this at your own risk. I assume no responsibility if this doesn't work for you. (Sorry for the lawyer speak but you know how it is)

Required items:

1) Soldering iron/solder
2) Volt meter
3) Philips screw driver
4) 8mm socket
5) T-25 torx driver
6) Electrical isolation tape
7) Some thin cable

You can do this before or after you've made the change described in the Heated Mirrors (Wiring to Rear Defrost Switch) article.

If you're not sure if you'll have this problem, just pull fuse #9 and switch the ignition on. The gear selector/temperature display should not be visible. If so continue reading.
First, you need to remove the instrument cluster. I followed the instructions here. One word of caution. The plastic cover on top of the steering column is painted. You can scratch it trying to remove the cluster (ask me how I know). To be safe, cover this area with a towel or something else first. Also, the left and right side connectors are a little tricky to remove. I used a screwdriver between the the connector and instrument panel and slowly rotated it. Once the connector is loosened, it removes easily.

After the cluster is removed we need to verify the wires we'll cut. You should have disconnected three major connectors from the instrument panel (I also had two smaller connectors). The ones we're interested in are the black middle connector that powers the temperature/gear selector display and the blue connector from the right side of the instrument panel.

Get out your DVM and set it so it can display at least 12v. Grab the middle connector and hold it facing towards you so that the wires are on the right and insert a small straightened paper clip into the last hole on the top row. Turn on your ignition and measure between the paper clip and ground. You should measure a little over 12v. Turn off the ignition, pull the #9 fuse and repeat the measurement. You should measure no voltage meaning we have the wire that's connected to heated mirror circuit. Turn off the ignition. If you look at the bottom of the connector where the wires come out, you should be able to tell what actual wire this is. On my car it was a black wire with a red stripe. Replace the #9 fuse.

Now, grab the connector that was connected to the right side of the instrument panel. Hold it so that the wires are to the right. Stick your straightened paper clip into the fifth hole from the right side of the bottom row. Turn on the ignition and measure between the paper clip and ground. You should measure about 12v. Turn off the ignition, pull the #9 fuse and repeat the measurement. This time you should still measure 12v meaning we've found an alternate power source for our gear selector/temperature display. Turn off the ignition and replace the #9 fuse. Looking in the bottom of the connector you should be able to tell what wire connects to this pin. On my car it was again a black wire with a red stripe.

What we're going to do now is cut the 12v wire from the black middle connector and splice it to the 12v wire on the blue right side connector.

First, get out your wire cutter and remove some of the black tape that covers the bundle of wires leading to the center connector. Trim it back so you have 4-5 cm of wire visible. Now cut the black wire with the red stripe leaving about 4-5cm of wire to the connector. Strip off some of the insulation.

Now take the blue connector and burn off some of the insulation on the 12v wire (black with red stripe) you found above. Don't cut this wire! Also, be careful not to remove any insulation from any other wires. I burned off about 1cm or so.

Cut a length of wire to span the distance between the two connectors. I cut a length so that I could lay the wire next to the bundle that runs between the two connectors. Solder the wire between the cut wire on the middle connector and wire with the burned insulation. Wrap some insulator tape around the two exposed ends.

Now carefully, lay the instrument panel on the top of the steering column cover. Reconnect all connectors. Turn on the ignition. The temperature/gear selector display should come on. Turn off the ignition and pull the #9 fuse. Turn it on again and the temperature/gear selector display should still work.

Tidy up the jumper wire by fastening it to the middle connector bundle with some insulator tape. Put the instrument panel back it and proceed to rewire your heated mirrors. Now you'll have the best of both worlds!