When in florence (3x) we always stayed at a Convent. It is very close to everything. It is right on a large square.The bus is right around the corner and takes you anywhere.
You can even walk over the bridge into the the major part of the city.
The cost is quite inexpensive for florence.
I wish I could remember the name. We haven't been there in 4 years. But is was always spotless. You can share a bath or get one in your room. We always shared and it was still spotless.
They do not have elevators so if you have many suitcases(we only travel with 1) that might be a problem.
One other problem is that most of the nuns speak only italian. HOWEVER they usually have someone at the front desk as a "helper" who is not a nun who will speak English. This is the person you will communicate with when making reservation.
there is a lovely garden out back you can go and sit in as well.
It if furnished in old antique sort of furniture but not shabby furniture by any means andt the beds are comfortable. Breakfast in the AM is sufficient but not excellent.
If you would be interested, I would try and find the name for you. I am sure I can by googling Convents in Florence.