Part numbers and (very) brief instructions w/pics (edited to add link)
Parts:
Mounting bracket (w/motor) 8D1 858 532 L 3FZ
Mirror shell 8D0 857 508 GRU
Mirror glass, heated, right 8D0 857 536 M
These are from the later A6- I believe Igor used A4 parts. YMMV.
Here is a pic of the stock RH mirror

And here are all the new parts

Basically, you need to take everything apart and swap out the stock short arm that the shell and motor mount to for the long arm from the new assembly. The tricky parts are getting the mirror glass off, extracting the wiring (and putting it back together properly) and reinstalling the retaining clip against considerable spring pressure.
First, take off the bottom of the small mirror, and remove the mirror glass. This is one of the fiddly bits- you have to be carefully pry the ring that retains the glass to pop the mirror loose.

Once the glass is out, you can remove the shell (1 or 2 screws). The old arm assembly is now in the same state as the new.

Then remove the power mirror motor (3 screws) to finish the initial diassembly.

Unfortunately, I didn’t get any useful pictures from here on, but you need to pull the clip that retains the spring from both arm assemblies (WEAR EYE PROTECTION). At this point, the arms will be loose on the pivots, and you have to extract the old motor from the old arm and the new motor from the new arm so you can swap the new arm onto the old base. The heated mirror connectors will pass through the hole in the arm, but you have to diassemble the connector for the power mirror motor to get the motor free of the old arm and onto the new.
The full instructions are available at: