What should I do about a female teacher whose hair repeatedly escapes from her khimar despite my having alerted her?

If the female teacher is in a place where no non-mahram men can see her, then there is no harm in her uncovering her hair in front of women, and it is not permissible to object to her. However, if she is in a place where non-mahram men can see her, then it is not permissible for her to uncover any part of her hair, and your drawing her attention to this is correct. It is incumbent upon you to advise and counsel her, and to repeat it if you do not fear any harm, in accordance with the Almighty's saying: "So remind, if the reminder should benefit." And make your advice good, and be a sincere companion, pure in intention for Allah.