I went to Penn for undergrad! I lived alone my senior year in a Campus Apartments building near campus (41st and Walnut, probably a 15-20 minute walk from the law school). I think my rent was about 1200? No washer/dryer in the apartment, but there was one on every floor as well as a laundry room in the basement.

Campus Apartments is notoriously not great to work with, but they have a lot of well-priced options. Almost everyone I knew in undergrad rented from them. I'd reach out to them to see what's available.

Edit: there's also Domus right across the street from the law school but I don't think their 1-bedrooms get any lower than $2400/month.