If a woman says the dua: فإن حبسني حابس فمحلي حيث حبستني During the intention to make umrah and she experiences her period during the state of ihram, does she have to give the penalty of exiting ihram?

No, there is nothing on her. Because she made the stipulation at her ihram, Imam Ibn Baz and Shaykh Ibn Uthaymin both held that menstruation is the obstacle she stipulated for, so she may come out of her ihram with nothing due from her, no hady and no fidyah. If she is able to wait, the better course is to stay in her ihram until she is pure, then do tawaf and say and cut her hair, and the umrah is complete.