We enjoyed the Hotel Royal at Via delle Ruote in Florence. It was formerly a well to do residence, so it has a lot of those attributes, like a lovely back garden, and a really pretty dining room where breakfast is served. It's in your price range and does have a/c. I can't remember whether there was an elevator or not.

Hotel Europa was very nice, a nicely furnished huge room for the 4 of us, the only bathtub we saw in our 2 weeks of travel, free internet, and a very helpful front desk. It's just north of the Duomo a few blocks, and about a block south of the Accademia. And the price was excellent, 180 euros for a quad. suite.

Pat