If two signatures is on the lease, it is a valid lease.
Yes, it is custumary to give a month's notice. 30 days ahead of you leaving, not within the last thirty days.If you knew the lease was expiring, why did you not put it in writing, and let her know you will be moving? It is always best to give as much notice as you can. Your post here does not give clear information. Was this a month to month lease, a six month lease, a one year lease? Refer back to the lease.