If the laundry room is at the back of my room, I can't sleep because I hear them talking all night, is this illegal invasion of my privacy?

I don't think it is an illegal invasion of your privacy. However, it may be a breach of the covenant of quiet enjoyment. You are entitled to the peaceful and quiet use of the premises you are renting. If the interference is bad enough that you must move out, you could have a claim for construction eviction. Such a claim usually requires that you actually move out first, though. I would talk to your landlord about the issue first, see if quiet times are being observed. If not, I would contact a local attorney for further assistance.