What is the ruling regarding a bedridden mother's inability to cleanse herself after relieving herself, and how does this obligation cease for her?

There is nothing against your mother — may Allah have mercy on her — concerning her prayer with the presence of impurity, as long as she was unable to cleanse herself and could not find anyone permissible to see her 'awrah (intimate parts), such as her husband, to help her. This is because Allah Almighty does not burden a soul beyond its capacity, and in this situation, prayer with the presence of impurity is permissible.