I "rebuilt" mine due to same situation as yours, but it's been awhile so my memory is a little froggy. Most likely someone will correct me if needed.
Inside door panel front corner is triangular interior piece - slide it straight up. If you pull it away from the door you'll either bend it and / or break retaining tab.
Remove screws [ 2 or 3?], disconnect electric cable, remove mirror.
Mirror housing has front cover and surrounding bezel [3/4" wide]. Carefully pry the front cover and bezel apart. In typical MB fashion, it's interlocking plastic tabs. Swap parts and reinstall.