Your tenant has trained you well.When you have an illegal tenant in a unit, you can send the resident a lease violation notice stating that the person needs to be on the lease, and they can pay the application fee for the background check.  

She gets the notice from the lien service and all of a sudden she is willing to pay 100% including the extra $600 in fees the lien service tacked on. 

Do I need to send her a notice to quit and have her sign it?

They were served 2x with notice of building being sold 60 days prior and 30 days prior as well as served a request to vacate by July 1st by a 3rd party.

I gave the tenant a notice to vacate.

Since they were all month to month, I provided the proper written notice, and followed all the rules, and eventually cooler heads prevailed, the tenants all moved, and I provided a free moving truck on moving day.

Your tenants will never notice the difference.

I do want the unit and don’t mind the tenant but I did not agree to take on a tenant with the hassles of starting  a 90 day eviction notice and let alone the rent is for less than I would have accepted.

Give them notice for violation of the lease.

@Charles Heistand I would NOT give month-to-month tenants notice in July of a rent increase in January so they can "prepare" for it.