Klab Conti - Just go there, ask to show you the place and the possibilities they can offer, you will leave with a subscription! Seems strange, but it seems they have all you need.
Check also the other 2 branches in Florence:
Klab Lulli and Klab Marignolle (amazing open air swimmingpool).
Via de' Conti 7, Florence-Centre, 50123View Larger Map