Praise be to Allah.
The majority of fuqaha’ are of the view that the one who does not pray for a while after he reaches puberty has to make up the prayers that he missed. If he does not know how many prayers he missed, then he should do what he thinks is most likely the number of prayers that he missed.
It should be noted that the prayers that she missed when she was young, before reaching puberty, do not have to be made up, because she was not accountable at that time.
Some scholars are of the view that whoever deliberately misses prayers does not have to make them up; all he has to do is repent and do good deeds in the future.
Anyway, what is required of your mother is to seek forgiveness a great deal, repent and do naafil deeds in the hope that Allaah will accept her repentance. Allaah says (interpretation of the meaning):
“And verily, I am indeed forgiving to him who repents, believes (in My Oneness, and associates none in worship with Me) and does righteous good deeds, and then remains constant in doing them (till his death)”
With regard to her saying that she made a promise to Allaah to make up the missed prayers for as long as she lives, this is a vow to make up the prayers that she missed, and she has done that and fulfilled that vow. So she does not have to continue making up prayers that she has in fact made up, because a prayer can only be made up once.
As for her saying “as long as I am still alive”, it seems that what is meant by this expression is that she will continue to fulfil her vow as long as she is still alive, and she will never stop making up prayers because of sickness, being busy or any other reason that may distract her from praying.
If she wants to carry on praying, this is a good deed that was encouraged by the Messenger (peace and blessings of Allaah be upon him) when he said: “Prayer is the best thing prescribed, so whoever can offer a great deal of it let him do so” (narrated by al-Tabaraani in Saheeh al-Jaami’ (3870)) – so long as that is done with the intention of offering naafil prayers, not with the intention of making up missed prayers.
And Allaah knows best.