I've been having a hard time finding a convent in Rome in September. I found this place:
They are near the Vatican and seem to have a lot of rooms.
They are also near the Lepanto Metro stop so it seems their location is not too bad (although it's not in the historical center). Their rates are reasonable...42 for a single, 77 double.
Their rooms look rather hospitalish but I guess that's all part of the experience, eh?