When the nanny is looking after the child, the child is the the responsibility of the nanny, not the parents. The kid wasn't there for a social visit (if so then it's almost certainly the parents responsibility even if they are not present)
But overall, the child's *behaviour*, as an entity in and of itself, is the responsibility of the parents. The ipad was broken as a direct result of that child's behaviour. Therefore, they should be responsible for paying for it.
If the child was unsupervised, I would totally agree. If my child got out of the yard, ran next door and broke a planter of the neighbours, I would pay. If I was visiting someone and my child grabbed an ipad and broke it, I would agree. If I left my child for a play date with a friend and she broke the ipad (deliberately) I would totally agree.
But, to my mind, all children have streaks of dreadful uncivilized behavior that they will demonstrate. Childhood is a bit of a confusing and crazy time for kids. That's part of the reason they behave like dreadful Mcdreadfuls sometimes. It isn't that they are bad. But they are learning civilized behavior. One of the very, very frustrating things about children is they don't progress in some linear slope towards adultness. You can have this fantastic two year old who says please and thank-you and is as sweet as pie who one year later morphes into a tantruming mess of a three year old who can become the epitome of a rage storm over a miscoloured soup spoon. How you deal with the behaviour determines your skill as a parent, but judging the end product alone is insufficient.
All this is to say that the nanny knew the kids behaviour and was responsible for the behaviour of the kid under her watch. The parents are not responsible for the "bad" behaviour. Some kids are wired differently and the best parenting in the world may not be enough to prevent violent outbursts.
So for unpredictable four year olds, the person in charge is responsible for the behaviour.
I don't think the knife anology is a bad one at all. There was something around that he should not have been able to access and damage resulted.