At 20 i had worked my backside off for 4 years while still at school and upped sticks to Australia on my own. It never occurred to me i could possibly get to the other side of the world and freak out (i actually freaked out 3 months later when i landed back in England and hated it!)
Put yourself in her shoes - she's young, never been away from home and suddenly she's in a foreign country, alone, no family or friends, no places really to go and chill or find her feet, probably not the foggiest how to look after kids..... How would you feel in that situation? She probably thought it would be brilliant and never thought she would hate it.
As for the dairy thing... I have a severely intolerant (not as bad as allergic, but the reaction is projectile vomit, cramps, projectile poop, face rash...) probably being a nanny isn't the best choice of job, strange choice on her part, as there tends to be yogurts, cheese, milk, ice cream pretty much daily and as a mum managing a dairy free 5 year old - dairy is in bloody everything! Even things that dairy really shouldn't be in.
Also as her employer that puts you in a bad place as you cant ask her to be around any dairy. Probably should have asked how serious it was before taking her on?
She's only human though - treat her with some heart. Yes it's a pain in the butt, but the poor girl is affraid and homesick. Imagine that's your child, away from home, alone and affraid - how would you want them to be treated? She's not just the hired help, she's someone's daughter, she's young and vulnerable and you are being pissed off she isn't looking after your kids (that's supposed to be what mums do, not young girls from abroad who are total strangers)