Yes I understand the 220 was the cost of the mirror, but if thats caused by the carwash.... it's an expensive carwash
I use a hard brush like the blue/white one in the picture:
Just gently brush the roof
But this type of 'deep cleaning' I only once did (4 years ago!), and my top is still clean with just the drying of the microfibre towels (car is garaged though)
It's especially good against algea (found in colder/wetter climates), and deep settled dirt. If your top is clean, and you don't park your car under trees (never do that!) or is parked outside in a wet climate it should stay clean forever. Just the drying with the microfiber towel is enough imho. What I always do is (when I dried the top, which is the last thing I do when I washed and dried the car), is open the top half way, and also dry (with a microfibre towel) the piece of roof under where the trunk seals to the top. That stays wet otherwise, and now you can reach it. That is a spot where algea grow fondly (i live in a wet climate).
The top is the weakest part of the car, and the less you wash it, the better it holds up.
When you decide to use the top weather proofer (to make it hydrofobe), clean it well (as dirt gets sealed in too).
But there are a lot of carcleaning/detailing forums out there with specialists.
For cleaning the top (before waterproofing), there are special ragtop cleaners, but imho any interior cleaner that is safe to use on doorcarts etc should do. They are all mild cleaners compared to soap based cleaners.