I might be missing something here but why are people suggesting you apologise to your daughter?
I have 4 children, all in their 40's now. There was only a 4 year gap between the first and last and all were totally different. I tell this story to show that I fully understand children misbehaving to the point of being out of control. But they can be brought back. Our 3rd was really hard work - biting, temper tantrums etc., though they were all brought up in exactly the same way, by the same group of family/friends, who all dealt with his behaviour in the same way. He was told a firm 'no' and what he'd done wrong. Until one day at the age of 3, in Tesco, he pulled a sweet stand over and threw himself on the floor face down, screaming and kicking. He had a thick duffle coat on. This unusual calm washed over me, I looked at him, asked a lady behind me to keep an eye on the other 3 then picked him up by the back of his coat stood him on his little stamping feet and walloped him on the bottom. He didn't even feel it through his coat but knew 'something' had happened. He just stared at me, speechless. I thanked the lady, gathered the four silent children and left the shop. (I rang later to apologise for abandoning my trolley!) Nobody spoke all the way home. Until my husband came home. Then, still calm, I told him he had the 3 as number 3 (he has got a name!) and I needed a chat. We went in the garden and all I said, very firmly was 'Don't you EVER do that to me again. Not at home, not when we're out. Not ever. Do you understand?' His little lip quivered and he nodded and he never did have another tantrum. He's grown up into the funniest, most caring human being and wonderful Dad (and son) in the world.
It seems the problem in the family here is that everyone is afraid of this child. Except Grandma who did the right thing, but then allowed her daughter to undermine her. Mum probably feels guilty because she knows what an absolute horror this child is. But the family needs to get a grip before she grows into an unchecked, uncontrollable teenager. All the child has learned from this is that her bad behaviour pays off. She can make adults argue and fall out.... what power!