"I think because the Nanny was the responsible caregiver, the parents shouldn't pay. This wasn't a playdate or a one-time babysitter. This is the nanny who should and does know how Herman behaves."
I'll double this back on my suggestion to be pragmatic about it. Consider how many nannies would continue watching him after being told that the hundreds of dollars in damage that he just intentionally did to someone else's property isn't his parents' responsibility. Consider also that Herman has a violent history and is destructive to his parents' property as well. Given these two things I would imagine that they won't have to worry about paying for any caregiver's property that Herman destroys because they'll quickly find themselves unable to find a caregiver willing to risk it. To my way of thinking, not paying for this kind of damage is tantamount to firing the nanny because it'd be a rare person who could take that kind of hit and keep doing the job.
"As Dindrane pointed out, the caregiver has responsibility for those behaviors the caregiver can reasonably influence. Putting away a small portable expenisve item would be somethign easily influenced to prevent the distructive action. I see leaving an iPad on a coffee table where young children can access it without appropriate monitoring as very careless."
This wasn't a situation where the child got hold of the device and damaged it while fiddling with it or spilled something on it, he grabbed it and deliberately threw it down the stairs. Given that, it's entirely reasonable to think that if the iPad wasn't there he'd have grabbed something else and destroyed it instead. Where's the limit? Also, the change of venue was entirely due to Herman's parents, and the nanny relocated him as a favor to them so again, that puts a lot more on them. After all, he wouldn't have been anywhere near the iPad if they hadn't had to send him out of the house due to construction. And, as I just said above, is it going to be worth the chance that they'll have to replace their nanny?