Dorms generally require that all occupants be officially listed and paying for living in the halls. Each university has a risk management department, and part of their job is to ensure that everybody in classes, at lectures, in the dorms, actually should be there. So many problems arise when unregistered guests remain in the dorms; thefts, assaults, arson and vandalism and more have been caused by unregistered guests. This is why most dorms have restrictions on the length of time that guests may stay consecutively in the halls.
I'd simply say, "Friend, I hadn't thought about this before -- of course, you're not the type to do these things, but my having you here is inconsiderate of fellow dorm-livers, and it's also seriously against the rules. I can't risk losing my housing spot, and I need to ask you to leave." And that's not random, exaggerating, or whatever -- it's the truth. Unless you can afford to be booted out and immediately on the market for a new place?
I'd try to get her out within a few days -- I imagine that that will make you vastly uncomfortable. So a compromise would be the day after the summer session ends, no argument.