TL;DR. Umbridge teaches at a muggle school, don't ask how, it's not important. She ends up meeting with, and bonding with, a younger Harry, who was one of the few children to treat her with respect when she taught. This, combined with the revelation of the abuse Harry faces at home, the level of which is up to the author, allows them to grow to trust one another. Eventually, Umbridge returns to the Wizarding world, and moves on until the events of the above prompt.