Only months later I was again confronted with a dangling mirror; this time probably an accident. I decided to have a try myself. I phoned around to some local car scrappers, but none had the right type of mirror, so I ended up buying another new one. It's a bit silly: a 19 year old car with the newest possible mirrors, but well, what can one do.
On Friday I googled how one replaces a wing mirror. Looked simple! On Saturday I bought my mirror. On Sunday I fitted it. It is quite easy! Even though it's awkward; the padding of the door, and the steering wheel, and all sorts of other car parts get in the way. But I have a mirror again, that doesn't need tape to stay in position! I did keep the old one; thanks to the tape, it is able to stay up. One never knows when the next helpful citizen thinks my car makes an excellent recipient for random frustration. I hope it won't be soon...
Step 1: getting the plastic cover off.