A disclaimer: Even though I feel compelled to leave you with my two cents, I am a new investor, with no experience managing or owning rentals. You have every look right to ignore me. Even so, I do hope you will consider my words.
It is clear that you have no personal obligations to this woman. She is not a friend or a family member, or an acquaintance from your past. The only relationship you have to her is through the document you both signed stating that in exchange for the privilege of staying on your property, she is required to provide you with x amount of dollars paid by date y of each month.
Though her relationship to you extends no further than the terms of that contract, it seems she has done more for you, through her exceptional care for her residence, your property, and her efforts to find you other good tenants.
My suggestion: Do more for her.
Be reasonable, and protect your interests, but please have compassion.
There are plenty of charitable organizations, churches, and other non-profits, who would gladly write a check to you for allowing this woman to stay in her current home, or provide her with a sum sufficient to move elsewhere. Call a few, and see what you can help her accomplish. And, though at least one person has suggested you move toward eviction, you and I both know that having such a recent mark on her rental history would make it difficult, if not impossible for her to find another place to live (not to mention the burdens such a stressful event would impose on her failing body).
She is a good tenant who has, for at least a year, contributed to your financial security and positively impacted the quality of your life. While I won't suggest you support her yourself, you are in a position to help her in a moment of desperate need. I don't see why you shouldn't.
