Suggestions in Florence and area

I can't help with specific apartment suggestions, but I assume others can.

However, I would suggest that you consider possibly splitting your Italian time between two locations since you do not plan to drive. Three to four weeks with a lot of day-trips by train and bus will pretty much limit you to Florence itself, I think, and even then you'll be going back and forth on the same rail lines multiple times and making connections (which can mean fairly long travel times). I think you'll run out of quick-and-easy side-trips long before your your 3-to 4-week stay is up. Obviously, if you're thinking of mostly staying put, just taking 1 or 2 trips away from your base each week, that would allay my concern.

I think you would probably enjoy spending some time based in a smaller city or town, though, and I suspect that would not be a practical choice for your entire stay because of transportation logistics. Many of the trips you took would probably start with a trip into a transit hub like Florence.

You might play around with one of the rail websites (I find the Deutsche Bahn easier to use for preliminary planning than Trenitalia) to see where you can get within 60 to 90 minutes from various possible bases. Start by Googling "rail map Italy" to see where the rail lines go.

Although I find its details concerning fares and travel times often far off-base, Rome2Rio.com is pretty good about identifying whether you need to go beyond trains (i.e., to buses) in order to get from Town A to Town B. If it suggests a bus, you usually can find a link to the appropriate bus company by drilling down. Sometimes Googling "bus Florence to San Gimignano" is faster, once you confirm that a planned destination doesn't have rail service or has service only to an inconveniently-located station.

Without a car, with one base only, the choice can be only one: Florence.

I've had good luck finding apartments in Florence using VRBO/Homeaway, although I can't make a specific recommendation as I haven't rented in your preferred area. Do a search for your desired dates, add any additional filters (e.g. max price per night, min number of bedrooms) and then scroll down to the bottom of the pictures and click on the Santa Maria Novella neighborhood. That will give you available rentals close to the train and bus stations (they're within a block of each other), but be aware that it may display some apartments a bit further from the stations than you prefer. Some landlords offer discounts for weekly or monthly rentals, and that can sometimes make an otherwise expensive apartment affordable. You may need to look at descriptions and pictures to determine whether an apartment has the outdoor space you desire. Pay attention to what floor the apartment is on if that's important for your father, keeping in mind that the first floor is one flight up. A washer, iron & board and free WiFi are other amenities you may want to look for. For me, those are necessities for a longer rental.

Airbnb is another source of Florence apartments, but I find the website much less user-friendly in terms of filtering for a desired neighborhood and apartment features.

EDIT: You can use the same filtering method I mention above to look for apartments in the other neighborhoods Roberto noted below.

Make sure you find out the exact address. Then come back here for advice.
The historical districts (quarters) of Florence are 4: San Giovanni, Santo Spirito, Santa Croce, Santa Maria Novella. Although the station is in SMN, however nothing is far, as the entire historical center is small. However landlords don't necessarily advertise by those historic names. They might say near San Lorenzo, or near Ponte Vecchio, or San Frediano. Being in the historical center (anywhere) would be my preference, not necessarily Santa Maria Novella.

I am in Florence now. We have an authentic remodeledone bedroom apt. We paid $600 Canadian for 4 nights. Our windows (open with shutters - honestly - right out of a movie) open onto the river. The ponte del veccio is a pleasant ten minute walk which keeps the majority of the crowds at the other end of the street. There is a double locked securitu entrance which from the street looks a little shady but is absolutely brilliant inside. You get a key. Kitchen, bath off of the bedroom, livingroom dining room furnished in white leather and antiques. King size bed. Really comfortable. 12 min walk to the train station or an 8 euro taxi. Super nice manager. Supermarket, laundry, and great restaurants everywhere. Message me if you would like photos. I will be back and I am pretty picky. 😊

Sorry for the typos. 😊