After a night of heavy drinking I've had a migraine that has lasted a week now, what I the problem?

Headache. This may be more than a migraine. For example you may have fallen and injured your head and you don't remember. You should consult with your doctor for an examination as to the cause of your prolonged headache.
Don't drink! You should see your doctor for any prolonged headache. Obviously, heavy drinking is not safe and should be avoided.