Notice period to vacate if rental agreement doesnt exist yet

Hello SIr/Madam, we havent signed the rental agreement yet. The tenant is yet to pay advance we agreed in full and that is the reason for the delay. Believing her, we allowed her to occupy the house and it is more than a month that she has been dodging to pay the remaining advance. She delayed paying the rent, maintenance as well and with great follow up we received the rent for June month 10 days. We are now skeptical about her continuing since she doesnt pick phone calls, makes false commitments on payment timelines etc. With not even a month of occupancy, since she created so much issues, we have asked her to vacate by this month end. She is now saying, we need to sign the agreement and she can vacate only after 3 months. Our concern is she will continue to create problems by not paying rent for the balance months and may also refuse to physically move out. My questions are : - Can we ask her to vacate with 1 month notice even without the agreement? - What should we further do legally if she refuses to vacate?