While I've never seen the workings of the outside mirror on a Chevy Sonic, I do have quite a bit of experience with OEM parts in general. That said, I've never seen a 'mechanical' remote-control door mirror that was designed to be repaired. They're simply remove-and-replace. And yes, you do have to paint-match them.
I would tell you to try buying a used one in the same color from a recycle yard, but most of them don't want to remove the mirrors from otherwise-good doors.
The alternative is do it the old-fashioned way and just push on the mirror glass until you get it where you want it.
Sorry to be of so little help...
Dave B.
Edit: You could also try searching for a used mirror on-line from a site like:
http://www.car-part.com/
These sites link dozens of recyclers and you could specify that you only want a particular color. If a recycler had a damaged or otherwise-unsalable driver's door, they'd be a lot more likely to be willing to take the time to remove the mirror.