My house has a separate apartment that I rent out. In my newspaper ad, I specified NO PETS because I am severely allergic to cats. The apartment and my living quarters share the heating system, so any hair makes its way into my rooms. Now my tenant has brought in a cat and refuses to get rid of it. My allergic reactions are making my life miserable, and my doctor tells me I must get away from the cat hair and dander. I'm desperate -- this person has a year's lease!
Giving up a cat is never easy -- but cat allergies are no joke. Your tenant was fairly warned. You made it plain in your ad that pets were not allowed, so she was legally bound by this rule unless you did something that would make her think you'd changed your mind.
If, for example, you've been openly tolerating the cat's presence, you might have a difficult time convincing a judge that you ought to be able to enforce this rule. From the sounds of things, though, it appears that you've acted promptly to remind your tenant that she is violating the "No Pets" term of her rental.
A violation of an important term or condition of a rental is grounds for termination. (Of the rental, that is, not the tenant; and certainly not of the cat.) You'll need to send your tenant a written notice telling her to cease the violation (as in, find kitty a new home) or move out. In legalese, this notice is called a "Cure or Quit" notice. If she does neither, you may file for eviction. Most states' laws give tenants a set amount of time (between three and ten days) in which to comply with the Cure or Quit notice.