What is the ruling of Islamic law regarding the fasting and prayer of a girl who suffers from an irregular menstrual cycle and continuous bleeding, and what are the implications for the days of Ramadan that she missed?

If the duration of bleeding exceeds fifteen days, you are in a state of istihadah (non-menstrual bleeding), and you must distinguish between menstrual blood and istihadah. Whatever possesses the characteristics of menstrual blood (its color, odor, and accompanying pain) is menses, and anything else is istihadah. If you cannot make a valid distinction, then observe six or seven days as is the custom of most women in your family, considering it menses. Then perform ghusl (ritual bath) and wudu (ablution) for each prayer after its time has begun. Brownish discharge (kudrah) is not considered menses unless it occurs during the accustomed period or if it is connected to blood. You must make up for the days you did not fast and the prayers you missed, thinking you were menstruating. Menstruation does not prevent one from remembering Allah, making supplications, sending blessings upon the Prophet, or reading the Quran without touching the Mushaf (the physical copy of the Quran).