Chicago has several schools in the city limits...UIC, DePaul, Loyola are the bigger names. Northwestern is a little north of the city and accessible on the CTA. School year ends in May for most but it might end later at Northwestern. I have no idea if they rent out dorm rooms. There is a gigantic hostel downtown near the Loop...the
J. Ira and Nicki Harris Family Hostel. There's also some pretty decent hotels in the suburbs that are fairly cheap.