If you look at the law suit, the mother didn't not notice the girl was gone. She filed a missing person's report with the police. It's very possible she left while the mother was at work, or told her mother she was going somewhere else, figuring by the time her mother figured out she'd lied, she'd already be in NY.
I'm not saying that her suing the show is right, but she DID notice. And the producer's better be able to find that consent form, and prove it's the mother's signature, otherwise they're going to be in a whole other heap of trouble for not validating the consent.
Don't most shows require a guardian along with a minor, though, anyway, to avoid problems like this?
And yeah, the mother does have bigger issues if her child really is a sex addict.