Month to Month Lease Help!?

I have a month to month lease. I just gave my one month notice on March 13th. She left me a note of improper notice because it wasnt before the first of the month, and said that I have to be out by Apr 13th (one month) but I have to pay for ALL of april. Can I tell her that if I have to pay for the month of April that I want the apt for all of april so she cant have it. Or since I gave her a written notice am I stuck paying for a full month and only getting to stay half of it??? I feel ripped off!