It is EITHER one month's prior notice, OR payment of one month's salary in lieu of notice, NOT BOTH. You can choose to give her notice (in writing, and ask her to countersign to acknowledge that she had read it and understands it).
If I were you - her improvement has come too late. I will still fire her.
She is just trying to tell tales on your daughter so that you will not believe what your daughter says about her.