What is the ruling on the fast of one who intended to fast a missed obligatory day on Monday, then forgot and woke up after Fajr and remembered, knowing that she broke her fast for the rest of the day?

Fasting to make up for missed Ramadan days requires the intention to be made the night before, by agreement of the four madhhabs. So, whoever does not form the intention during a portion of the night until dawn breaks, their fast for that day is not valid. They must make up that day and form the intention for it the night before.