Ah the Art Gallery Hotel. I used to live on Via delle Terme, right around the corner.

There used to be a laundry place on Via Faenza (I used it) but that was back in 2005. Not sure if its still around. My friends tell me that, in typical Italian fashion, there's a place called 'The Laundry Cafe' (http://www.thelaundrycafe.it/) in the northwest part of the city. You can do laundry and enjoy a cafe all at the same time. They said the prices aren't reasonable but aren't steep, whatever that equates to.

Whenever I stay in a hotel I always ask the front desk what place they send their laundry out to. Surprisingly most will just tell you up front but some may ask why you want to know. Just tell them you'd like to check out reviews before sending your clothes out to them. Once (or if) they tell you then just go there yourself and you won't have to worry about paying the hotel's surcharge. Just a FYI, I've only done this in the U.S. so you'll be the guinea pig if you try it out. Please post back how the Laundry Cafe is if you try that place out too out. Have fun in Firenze!