How do I remove painted knockdown texture?
I don't know that replacing the drywall for that little a space wouldn't be the best route.  Seems the cleanest.  Removing trim isn't hard.  I don't know about the mirrors, though. Renting a drywall jack is cheap and makes it a ton easier to hang.  A bit of demo and hanging a new ceiling sounds easier than messing around with something that isn't working out.  

Beyond that, I don't know about knockdown texture, but sanding down a ceiling without a drywall sander sounds like it sucks and trying to water/scrape it sounds like it sucks and is super messy.  Demoing drywall is quick.

No expert here.  I'd just replace it.  Seems the surest, which is the direction I like to steer.