Get your insurance company on the case - if no insurance then your lawyer. Take photographs of damage. Get damage repaired. Seek compensation. Tell neighbour what you intend to do.
Water coming into your place from upstairs is the responsibility of the people upstairs - do not get concerned by their continuing to use machine - it's just incidental - the key thing is that it was their water - negligence, stupidity, deliberate or plain unhappy accident makes no difference. You have been damaged by their water - end of story (as they say).
Peccavi, October 2008