Here is a good option:
www.florencegrandtour.com

Thanks Bob, certainly worth a look.

Pete..try La Residenza..a winner on Tournobuoni Street, wire cage elevator, neat hotel, large rooms/bathrooms, rooftop views, breakfast. It was a great find for us. Stu

www.laresidenzahotel.com/

We have stayed twice at the Tourist House Ghiberti. It is right behind the Duomo. Rooms are modern and very nice. It has an elevator (rooms are on the 2nd floor of the building). The price is a bit higher than you want - about 150 euro per night but is a wonderful location and worth a look.

www.touristhouseghiberti.com

Take a look at Residenza Johanna 1.

It's just a few steps from the San Lorenzo market and the Museo San Marco, and some nice informal restaurants in a range of price categories.

http://www.booking.com/hotel/it/residenza-johanna-i.en-gb.html?aid=311984;label=residenza-johanna-i-4Noo2cg6GPXszF3rYB0eMwS13987270153%3Apl%3Ata%3Ap1%3Ap2%3Aac%3Aap1t2%3Aneg;sid=9688cd171d09c4509d2d0b4e08180a84;dcid=1

I meant to add that it is in your budget for April rates.

Hi here's a hotel right in via de' Cerretani, the street leading from train station to Duomo http://www.hotelperseo.it/. It is in a short walking distance from both and the staff can help you to book a visit to Uffizi museum so not to wait in line.
Unlike others hotels the hall is open 24/24 per day.

Hi, all these hotel are in the city centre, walking distance from Duomo and train station, right in via Cavour, which is the street connecting Duomo to Piazza San Marco):

http://www.hotelcasci.com
http://www.hotelcolomba.com/
http://www.webhoteleuropa.com/

I hope this is going to be helpful.

Thanks all for your suggestions - it certainly makes selection easier to have these recommendations.