Are the prayers my mother performed after the end of her irregular menstrual period, which lasted 15 days, considered valid, and must she make up for the days of Ramadan if she fasted in that state?

Scholars have differed regarding the maximum age at which a woman menstruates. The preponderant opinion is that there is no maximum limit. Consequently, your mother is experiencing istihada (non-menstrual vaginal bleeding) because the bleeding has exceeded fifteen days.

She should refer to her previous habitual period, and whatever exceeds it is istihada. If she does not know her habit, she should act based on a sound discrimination (between menstrual and non-menstrual blood). If she has neither a habit nor discrimination, she should consider herself menstruating for six or seven days, based on investigation, then perform ghusl (ritual bath). She should consider whatever bleeding occurs beyond that as istihada, and she should fast and pray during the days of istihada. She must make up the days counted as menses after Ramadan.