First off, I don't have any idea of what I'm talking about. I don't know you, I don't know your friend and I don't know the circumstances of your friendship. Below is from the feelings I got from your post...
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It's very kind of you to give the little girl nearly a week of your time despite all of your other obligations. I worry about her, if her mom's willing to smack her in front of you what does she do in private???

How important is this friendship to you? Is it really worth hanging on to when it seems to cause you so much stress? Is she really a friend if you'd lose her through a bit of hard honesty? Are you finding yourself feeling more doormat than door? Is there someone, maybe some sort of child abuse counselling place, that you could ask about aproaching this very sensitive subject? Someone who could help you find the words to use? Your kids' school counsellor maybe? Do you really want your kids to witness her behavior towards her kid? Do you really want to continue to witness it yourself? Can you NOT say something, even a tiny something??

Wow, I can't imagine how hard this is for you. Best of luck to you and to this woman's little girl and dog. Best of luck to your friend too, she sounds like she could use a little help in relationships with other living beings. The way she does things may be the only way she knows, maybe the way she grew up with herself? She really may not realize there is a different way of doing things.

Who ever heard of 'training' a guy through example?? And here I've been using the frying pan method! (ok, really, if I want my husband to do something he hasn't thought of I have to ASK him, then let him make his own choice).