I think for this, and for damage in general, there is a difference between an accident that occurs in the normal course of affairs, and deliberate negligence or destruction.
So if you have a party and someone spills a glass of wine on your nice carpet, it's your responsibility to have it cleaned - the occasional knocked over drink is a natural and expected consequence of entertaining. If your roommate is washing dishes and drops your favourite cup - again, a natural side effect of normal living conditions.
However, if you're having a party and someone gets drunk and throws a glass of wine at someone, or someone decides to play catch with your crystal, then they should pay for any damage that results because their actions are not within the normal bounds of expected behaviour.
In this case - if the nanny were normally babysitting in her own home, then I'd lean towards the fact that she should know what the behaviour of her charges is like, and that part of the setup for watching the kids would be some appropriate childproofing. She would also have the option to get insurance that might cover damage that occurs in the course of her job.
Given that she was doing a pretty major favour for the parents by looking after the kid in her own home, and that having an iPad on the table is a very normal thing to do in her own home, I'd go for the parents' paying for the damage.
If I were in the nanny's situation, and the parents' didn't offer to pay, I'd be looking for a new child to take care of, and I wouldn't worry too much about giving ample notice. I'd also be extremely reluctant to have the child in my home again - as others have said, now that he's trashed the iPad what's next? The TV? (LCD screens don't hold up well to heavy objects being thrown at them) Her computer? Maybe the lamps? I'm guessing that the nanny doesn't have the sort of space where she could remove all valuables to other rooms to give the kid a closed off safe room where he can't damage anything.