A lease agreement was never drawn up for tenant, but would like to have tenant to vacate property because she is unable to pay the increase of rent requested. She is aware that she has to move and has been given several option. She is okay with that but I want things to be done legally.

A lot of states require that you provide the tenant 30 days notice of your intent to raise their rent. If she is unable to afford the rent increase, then she needs to provide you with a notice that she is moving. If she refuses to move or to pay the increase after you have given her ample notification, file a request in court for eviction.