You do have a place to live, but I agree that you should at least receive a discount for your inconvenience. You should ask for a concession from your landlord and if you receive none, you could sue the landlord for the amount you feel would be justified under the circumstances. You will need evidence.
You should give the landlord notice and move. It sounds like the apartment is no longer habitable as it has a large gaping hole in it and is allowing the elements in.