good area to stay in in Florence -- and accommodation suggestions

Oltrarno (beyond the Arno, i.e. South of the river). That's relatively less touristy still (but getting that way) and a little cheaper.
Try hotel la Scaletta.
http://www.hotellascaletta.it
If they don't have availability, do a search on venere.com or booking.com, but stick to that side on the south side of the river Arno.
$200=€140, that should buy you a nice three star hotel in Firenze.
3 Stars in the Tuscany Regional rating system is very adequate. The Sheraton for example gets 4 stars but for a 4 stars $200 aren't sufficient.

Relais Grand Tour is a bed & breakfast about a ten-minute walk north of the Duomo. Quiet, safe neighborhood and a great value. I stayed there in March and highly recommend it. Check it on Trip Advisor.
Cynthia

We stayed at the Hotel Casci a few years ago and found it convenient and well priced. While the entrance is unprepossessing, the room and bath were very clean and nicely equipped. The management was very friendly, providing me with laundry service and an ironing board upon request.

I second the Relais Grand Tour. We stayed there in May. The location was wonderful. It was very close to the Accademia. The price was extremely reasonable. From what I remember, it was approximately 90€ a night. It is located in a quiet, safe area. The owner is helpful to point out restaurants and things to see. He was genuinely kind. We stayed in two of the rooms upstairs. The bathrooms were small (normal for Italy) but the rooms were a nice size. The ground floor has very nice suites.

Howell,

If you want to be close to the centre of Florence and sights like the Duomo, San Lorenzo market or whatever, you might have a look at THIS B&B. They also operate a nice Trattoria on the main floor, and there's a Laundromat right across the street.

I'd also recommend Katti House. We stayed at Katti 2 on via nationale, ask for a room at the back for a quieter night, though windows are all double glazed.

It was a couple of years ago but we liked our stay at the Hotel la Venere in Florence. The location was excellent, maybe ten minute walk with luggage from the train station, five minutes or so to the Duomo. Our room was comfortable, the bathroom was quite large. The staff were friendly and helped us several times, including arranging the Uffizi for us. Breakfast was fine, a charming young lady made it pleasant every morning. We got a freebie glass of wine each evening! It was certainly below the price point you note. We were quite satisfied, but we are far from fussy about what we require from a hotel. Have a wonderful trip to beautiful Florence.

Ditto on La Scaletta. Best, and terrace overlooks the Boboli Gardens.

We stayed at the Pensione Bencista, and it was incredible. It is a 10-15 minute bus ride from the city, but the views are beautiful, food is great, and it has interesting architecture (it was formerly a convent). It was one of my favorite places I've ever stayed, and we've traveled pretty extensively. It's definitely not your average hotel.