What is the ruling regarding a mother praying while seated for a long time, despite being able to stand, and what is incumbent upon the questioner towards her, and what must she make up for of past prayers?

It is not permissible for the mother to pray sitting in a fard (obligatory) prayer if she is able to stand, and her prayer would not be valid then, by consensus. However, if standing causes her extreme difficulty and harm, then it is permissible for her to pray sitting, and her prayer would be valid. You must advise your mother and explain the religious ruling to her. If she responds, then she must pray in the religiously prescribed manner from now on. As for what has passed, there is a difference of opinion among scholars regarding the obligation of making it up, and it is more cautious for her to make it up. If she does not respond, then you have fulfilled your duty, and continue to advise her and supplicate for her.