Is what a woman does—performing ghusl, prayer, and fasting after eight days of her menstrual period have passed, and considering anything beyond that as istihadah—correct or incorrect, knowing that she takes medications that affect the regularity of her period and the prolonged continuation of menstrual residue?

If the bleeding duration exceeds fifteen days, then you are experiencing استحاضة (prolonged non-menstrual bleeding), and you must refer to your previous habit (eight days of menstruation) and then perform ghusl (ritual bath). On the extra days during which you see blood, you must perform wudu' (ablution) for each prayer after its time has begun and then pray, taking precautions by wearing a tightly-bound cloth. If the bleeding stops at certain times, then you must pray during the time it stops. However, if the bleeding duration does not exceed fifteen days, then all of the blood is considered menstruation, and you must wait for it to stop, then perform ghusl and pray.