Native English speaker wouldn't say "wash the mirror" -- clean, or wipe, or polish maybe. The verb "wash" suggests dunking in a sink or running through an appliance (like a dishwasher), so if you say "wash the mirror" the hearer will think something odd is happening.
I need to get to washing the mirror was marked incorrect. Is it really wrong?
I suppose some people might say it, but I wouldn't recommend it to any learner as the correct phrase. It also has a slightly different emphasis. If you say: "I need to wash the mirror", it is neutral. If you say: "I need to get to washing the mirror" it sounds like a job you have been repeatedly putting off, or couldn't do because of other things.
Which reminds me: I need to "get to" doing the housework, otherwise I'll waste all day on Duolingo - AGAIN!
P.S. I know it's not really a waste, as learning is never wasted, but it doesn't get the mirrors washed - or anything else. ;)