What should the mother do to atone for swearing to her daughter that she would not wear contact lenses, especially since the daughter needs them and her eye was previously harmed?

If your mother wishes to retract her oath, she must expiate it, based on the Prophet Muhammad’s (peace be upon him) saying: "By Allah, if Allah wills, I shall not swear an oath and then see something else as better than it, but that I will do what is better and release myself from the oath." In another narration: "but that I will expiate my oath and do what is better." The expiation for an oath is to feed ten poor persons, or to clothe them, or to free a slave. If she cannot find any of these, then it is to fast for three days, as Allah the Almighty says: "Its expiation is the feeding of ten poor persons from the average of what you feed your [own] families or clothing them or the freeing of a slave. But whoever finds none – then a fast of three days." (Al-Ma'idah: 89).