We stayed at Suore Santa Elisabetta last September. It is walking distance from the Termini, close to St. Maria Magiore and a short walk to the Forum and surrounding sights. Their email is email@example.com. I found them in Rick's Italy 2007 book. There are many convents and monestaries listed on santasusanna.org and also other good travel info.
Santa Elisabetta's is run by Polish nuns but the front office was managed by an efficient, friendly, English speaking woman. They did have a curfew at 11pm, I think, but we were so tired from walking all over Rome all day that we were happy to be soaking in a hot bath by then. The room was comfortable and one of the largest we have stayed in on our travels to Italy. Pluses: great price, good though basic breakfast with excellent coffee, large private bath with a large clawfoot tub, comfortable bed, very clean, very good security, good location, quiet except for the sisters singing morning prayers at 5:30 or 6am [this was actually a bonus for me, a la "Sound of Music", and I fell right back to sleep thinking I had heard angels;) ]. Minuses: twin beds, short window of time for breakfast - if you want to sleep in you will miss it, but no problem if you want to get an early start, minimalist decor- it's a convent. We booked this rather late so there was only one room available, but I understand there are some rooms on the upper floors with great views and possibly a balcony. If you do a search on tripadvisor.com for them you will find more reviews and some photos. I think our stay with the sisters was an appropriate home base for our visit to Rome and the Vatican City.