This is a tough situation, and I honestly don't know what I would do with this tenant. Your best option may be to appease her for the moment and not renew her lease or do an inspection and note the cat pee/ vomit.
Put the liability onto them. Tell her you dont feel comfortable supplying the AC in her unit, as you dont want to be responsible for her health. ask her to go get one of her own and maintain it herself. Take a bit of the rent to compensate her for the monetary outlay, if the window unit cost 200 then reduce rent by 20 per month for 2 years ect.
Every tenant has been programmed to use the term "Black Mold" for all mold and mildew. That said, you probably don't have a mold issue but obviously an AC unit is going to generate condensation and mildew. It doesn't sound like a very appealing tenant so I'm torn on what to do but I guess simply because she mentioned the M word (which I never use around tenants) I would probably replace it and have that one cleaned up as a backup unit.