I stayed at the Della Signorina Hotel in Florence last fall. I comes in around your budget, and is very close to the Duomo and is just 3 blocks from the Ponte Vecchio and not very far from the Ufizzi. I booked through Venere, search Florence in the Ufizzi area.

I walked to the hotel from the train station, it was about 10-15 minutes, and the neighborhoods were fine. The staff was friendly, the continental breakfast was included and was decent enough to get started for the day.

I would stay there again.