I think both are at fault, though the girl moreso. The sitter is at fault mostly for not seeking medical care for the baby, but the actual act of violence is on the girl, possibly the parents, though i have a hard time imagining any parents with a new baby that don't explain to their older children that babies are fragile and you must be gentle with them.
Some of the comments in this thread baffle me, people who think a child of 10 can't understand death or that hurting a baby is a bad thing. While true that we don't FINISH development of our personalities, morals etc. until we are 20ish, it's not a light switch being turned on, it's a gradual progression. At 10 the vast majority of children do know right from wrong, and know violence is definitely not ok, especially towards a baby.
That being said, this is hardly the first time a child has murdered someone. http://karisable.com/youngmurder.htm Whether that's due to bad parenting or mental/emotional disorders, or something else, I'm sure it varies. But one thing is clear, many of these children, when questioned after the fact, make it clear they knew it was wrong, they just did it anyway. Are they clinical psychopaths? Or acting out after abuse? Again, probably varies from instance to instance, but they certainly did know that what they were doing was wrong, they just didn't care for whatever reason. I don't think this excuses their actions, lots of people have bad upbringings and don't murder people. You don't let that slide, if they knew it was wrong but didn't care when they were 10, unless immediate and strong treatment is given, and constant, thorough supervision, they will likely do it again.