The daughter is not the daughter of the speaker, but addressee, However the speaker is the one who raised her and perhaps didn't do a good job. It's really 娘を or not 娘に, so the way I understood it, was she turned out that way because of the speaker/or else, but the translation in English makes you think that perhaps the speaker did something else to the daughter. If I change 娘 for 宿題, would it be:
宿題をあんな風にしたこと : I did my homework that way
宿題にあんな風にしたこと : I did something like that to my homework
Doesn't make much sense, but it's just an example.