We stayed at the sister hotel of the Due Torri (owned by same beautiful Roman woman) called the Fontanella Borghese. The two hotels are very near each other, but I would pick the Fontanella Borghese's location... in the middle of a great little neighborhood that we enjoyed very much. I think its rate is a few euros more than the Due Torri, but both would be within your 225 euro budget.

I don't know for sure if this hotel qualifies but Hotel dei Mellini on the other side of the Tiber is a good choice. It's closer (but not right next to it) to the Vatican in the Prati neighborhood. It's on a very quiet street. We stayed a couple of blocks away at the sister hotel, Visconti Palace, and walked past the Mellini every time. And we wished we could have stayed there instead. Although the Visconti was good the Mellini looked much nicer.
According to a comment from tripadvisor.com, the hotel can give a better rate than what's shown on their website.

Many, many thanks to you all. I am relatively new to this site and am very appreciative of all of the insightful, helpful information here!!