Is my mother's prayer — she who does not utter a complete word, memorizes nothing, prays while sitting, and only offers the Takbir and moves her hand without knowing the number of Rak'ahs — accepted, and is there a solution for her condition?

If your mother has lost her mind, she is not obligated to pray, because accountability is lifted from one who lacks intent. However, if her mind is sound, she must perform as much of the prayer as she is able. If she is able to stand, it is not permissible for her to sit. If she can memorize Al-Fatihah, she must do so; otherwise, she should say: "Subhanallah, Walhamdulillah, Wa La Ilaha Illallah, Wallahu Akbar, Wa La Hawla Wa La Quwwata Illa Billah" (Glory be to Allah, and praise be to Allah, and there is no god but Allah, and Allah is the greatest, and there is no might or power except with Allah). If she is unable to do any of that, nothing is binding upon her, and she should perform what she can of the actions of prayer.