How does the mother perform her prayer, and is it necessary for her to be in a state of purity, knowing that she is paralyzed in the lower part of her body and cannot move from the bed to relieve herself except with assistance?

Your mother should fear Allah as much as she can and pray according to her ability, for Allah Almighty says: "So fear Allah as much as you are able." If she is unable to stand or sit, she should pray lying on her right side, facing the Qiblah with her face and chest, and gesturing for bowing and prostration. If she cannot do that, then she should pray lying on her back.

If she cannot control what comes out of her, then her ruling is that of one afflicted with incontinence; she should perform ablution for each prayer after its time has begun, and take precautions as much as possible. If she cannot find anyone to help her with ablution, she should perform tayammum for each prayer after its time has begun, and she must avoid impurities as much as possible. Whatever obligations of prayer she is unable to perform are waived for her.