Is it obligatory for us to fast on behalf of our mother for the days of Ramadan she missed due to menstruation and did not make up out of ignorance, given that we do not know the exact number of those days?

The deceased woman is obligated to make up for the menstruation days she missed in Ramadan. You are permitted to fast on her behalf, and the number of days should be estimated based on your best judgment if you do not know the precise number.