Rome is seldom very cold, even in winter it may offer wonderful sunny days in which walking is pleasant. Moreover, this month there are sales everywhere, so it's an excellent period for shopping. And there are plenty of museums at hand, just in case it rains!
In January Casa Cristina is cheaper than ever: only 110 Euros per day or 690 Euros per week. And, as usual, more discounts for long stays.
Hurry up!