You could have asked the estate to proceed with an eviction prior to your closing on the property. As it stands, you chose to accept the property knowing that the woman was staying on the property. Your notices prior to the closing don't amount to anything, because you weren't actually the owner of the land at that time and had no standing to give notice or commence eviction proceedings.

A cursory summary of the eviction process can be found here. Notice requirements are summarized here, and some sample forms for giving notice can be found here. Eviction forms can be found here (under Landlord). It may still be in your best interest to offer her something in exchange for her voluntarily vacating the premises.