Are the prayers of one who did not perform ghusl after menstruation due to her ignorance of the validity of the prayer, valid, and what should she do if they are not?

You erred by neglecting to learn what was obligatory for you. Your prayers, along with your failure to perform ghusl (definition) (ritual purification) from menstruation for an impermissible duration, obligate you to make them up according to the majority of scholars. You should make them up by estimating until you are reasonably certain that you have fulfilled your obligation. Sheikh Al-Islam Ibn Taymiyyah held the view that making them up is not obligatory due to your ignorance of the ruling, but the more cautious approach is to make them up, in accordance with the opinion of the majority.