I am a homeowner, I rented a room with use of common areas of the home to a tenant, I now want to end the living arrangement. She constantly pays the rent late and has recently damaged the property and isn't willing to pay for the fix of it. There is no written agreement and therefore I believe it is a month to month tenancy. What is the protocol for non payment and damage to the property. Thank you.

Written or not, you follow the landlord-tenant laws of your state. First give her notice to leave in writing! In most states you have to give 30 days. Florida is only 15. Calif requires 60 in some circumstances. Calif and some other states have 'lodger' statutes that allow you to terminate sooner (and she is a lodger, in those states only).
As soon as you have done that, start contacting your local courthouse for eviction procedures where you live.
You can evict for non payment of rent, and/or for damages. You can simply terminate the arrangement as well, with no reason stated. Even if she pays you rent to be current, that doesn't stop the eviction.