Is it permissible to sit next to one's mother during prayer to recite the Tashahhud to her, so that she can repeat it after me until she memorizes it, and what should be done if she refuses?

The Tashahhud (definition) must be memorized. If a worshipper is unable to memorize it or is pressed for time, they should read it from a paper or from what is written on a wall, provided they can read well. If they are unable to read, they should replace it with tasbih (glorification of Allah), tahmid (praise of Allah), tahlil (declaration of Allah's oneness), and takbir (magnification of Allah), while ensuring that this substitute includes the monotheism of Allah and praise for Him. Saying "Al-Tahiyyat Lillahi wa al-Salawatu wa al-Tayyibat" (All salutations, prayers, and good things are for Allah) along with the two Shahadas (declarations of faith) is sufficient for the Tashahhud if nothing else is possible. There is no harm in prompting the worshipper during prayer for the Tashahhud, or for any other statement or action they are unable to perform, even if it requires seeking help from a prompter for a fee; their prayer remains valid. If teaching and prompting are not possible, they should perform what they can of the prayer's statements and actions, as Allah Almighty says: "So fear Allah as much as you are able."