I have one rescue puppy too. She is the black and silver one in my profile.
They said she was house broken too. But a new place, other dogs, and we started over.
I took her out every hour, I went with her, commands were, "go potty now". then " good girl'. When she came in she got to play for about 30 minutes and then she was kenneled for maybe another 30 minutes. Then i repeated this. After a few weeks of being in the kennel, i started with a leash in my belt loop for the same amount of time.
This routine got to be such a habit for the both of us. When se became a year old i noticed I was still taking her out repeating the commands. I decided she was the one that needed to out not me.... but the habit was established. Now I send her out with the other dogs.....whew.
Looking at the look on your daughters face, mom, you have done a good thing. I am sure it will not be without trials. Kids learn a lot by taking care of an animal.