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Welcome to JustAnswer.com! My name is***** and I have been a licensed veterinarian for 10 years. I would be happy to answer your question. I'm so sorry to hear that Mia attacked your daughter. I hope she didn't break the skin with her bite. If she did, your daughter needs to wash it thoroughly with soap and water for a good 10 minutes. As far as getting Mia more acclimated, I recommend buying a Feliway diffuser. It plugs into the wall and releases calming feline pheromones. If she is able to get safely near her, your daughter can also try getting a Nurture Calm collar - it works the same way as the diffuser except the phermones are near her face all the time, not just in the air. Again, only attempt to put the collar on her if she stays calm enough to let you do it. If she hisses or growls, do not continue to try to put the collar on. Mia sounds like a cat that wants to get to know your daughter and her son on her own terms. I wouldn't try to pull her out to play or carry her anywhere. I know this is hard to do because they are so cute, but it is much better (and safer) to give them time to come to you. So for now it will be a one way relationship with kitty getting fed and taken care of. Once she starts trusting that your daughter and her son will provide consistent care (and that they won't eat her), she should start coming out of her shell and start interacting in a friendly way.
It really depends on the individual cat's personality. We don't know what her past history has been. She may have faced some trauma from human interactions in the past, or she could just be a shy and nervous cat. I would not keep her isolated and prevented from exploring the house as long as she doesn't initiate attacks. If she wants to look around the house and doesn't attack anybody, she should be allowed to explore. She will let you know if and when she wants to be petted by coming up to you. Trying to pet her if she doesn't ask for it just makes her anxiety worse.