Responses

Basically because the real daughter (forgot her name) asked to see her "mother" he decided to spare Katherine's life after consideration. Knowing that harming the royal family is a crime punishable by death, he called everyone who knew the switch was intentional in his office. He then told them that Philomel's mom didn't know that the babies were switched. Making her innocent so, she's escaping the death penalty.
I don’t get the whole father scene