My wife and I stayed here for 8 days total, 5 of which had NO WIFI access. We were told it was a problem with the provider. When I asked the host for a small discount of one night free to combat the charges we had to pay for purchasing a weeks pass of wifi each, she responded:
"I am Sorry for these first days problem, but it was really not under our control.
Given the extend of Your stay you have already had a 15% discount on the normal room price, and Infact you are paying 39 euro per night, that it is a very low rate for this period , the area and the apartment. I hope you will understand."
So basically: this was a nice enough, relatively inexpensive apartment that sometimes may or may not have internet access. Regardless of it being "beyond the hosts control", it is up to them to rectify these situations. We gave them an opportunity to so do and they shut it down with no acceptance of fault. So if anything does go wrong with your trip here, I wouldn't expect much help from the host.
Enjoy Firenze!