I agree with failing this student, here are some other advices.

1. As said by other answers, make sure you have collected e-mail correspondences in case if she files a grievance.
2. If you need to write to her, write as short as possible. Simply state the fact in a couple sentences. Do not put any evaluation or judgmental statements in the e-mail. The more you write, the more she can later pick on.
3. Once she missed the ultimatum, simply submit the grade as soon as you can. If she comes to talk to you again, tell her that it's been in the formal record and out of your control. Refer her to the registrar.
4. If possible, do not meet with her with just you two. Invite at least one supervisor, registrar, or other relevant person to be there. Desperate students can potentially make up a lot of delusional accusations; having a third person can avoid the "you said, I said" confusion.
5. Just give the simplest explanation, repeat that if you may. Even if she presses for more explanation or justification, always go back to the simplest explanation. And feel free to tell her there is no more you can say. If she explains with learning disability, accident, or hurricane, do not join the discussion. Say again the decision has been made and it's out of your control. Refer her to the high up.
6. Get the next in the line involved. It can be the Dean of Students or the Dean of the Department. Let them know you may refer her to see them if she requires to advance her plead.

Remember to chill out. Failing students is in no way a pleasant experience. However, it's our job to give objective evaluations. The student lacks fundamental cognitive input and basic professionalism; to me, it's a one way ticket to the Fail-land.

I agree with failing this student, and here is some other advice:

1. As mentioned in other answers, make sure you have collected e-mail correspondences in case she files a grievance.
2. If you need to write to her, write as little as possible. Simply state the fact in a couple sentences. Do not put any evaluation or judgmental statements in the e-mail. The more you write, the more she can later pick on.
3. Once she misses the ultimatum, simply submit the grade as soon as you can. If she comes to talk to you again, tell her that it's been in the formal record and out of your control. Refer her to the registrar.
4. If possible, do not meet with her with just you two. Invite at least one supervisor, registrar, or other relevant person to be there. Desperate students can potentially make up a lot of delusional accusations; having a third person can avoid the "you said, I said" confusion.
5. Just give the simplest explanation, repeat that if you may. Even if she presses for more explanation or justification, always go back to the simplest explanation. And feel free to tell her there is no more you can say. If she explains with a learning disability, accident, or hurricane, do not join the discussion. Say again that the decision has been made and it's out of your control. Refer her to the higher-up.
6. Get the next in the line involved. It can be the Dean of Students or the Dean of the Department. Let them know you may refer her to see them if she requires to have her plead advanced.

Remember to chill out. Failing students is in no way a pleasant experience. However, it's our job to give objective evaluations. The student lacks fundamental cognitive input and basic professionalism; to me, it's a one-way ticket to the Fail-land.