What is the ruling regarding the prayer of a woman with irregular menses who used to shorten her prayers out of ignorance; does she have to make up the prayers she missed?

You must make up for all the prayers you shortened during the period of irregular bleeding (istihadah), because shortening prayers is only permissible when traveling. What you did by shortening prayers during istihadah was a mistake, and your prayers are invalid because of it, according to the majority of scholars. You must count the number of prayers and make them all up, or make up what you predominantly believe will clear your responsibility. While the view that an ignorant person does not have to make up past prayers has some merit, the view of the majority is more preferable and cautious. Any blood discharge beyond the normal menstrual period is considered istihadah and does not prevent prayer.