It is highly unlikely that with a tutor, your daughter can learn enough Mandarin Chinese for it to be useful (I'm telling you this from personal experience). A language with a Latin alphabet is probably a more realistic option. Although, really, making use of a language learned with a tutor, is pretty much impossible. Which doesn't mean she shouldn't be learning. Learning foreign languages is great for overall brain development, etc., and it could certainly give her a leg up if she ever finds herself among its native speakers.

I studied Spanish in high school and college, and it overall didn't turn out to be particularly useful (within the past year, it has helped when communicating with resort staff in Mexico, and with painters/floor guys doing renovations in our house). Spanish is easier to learn than French (I've tried both), but wanting to learn a language is considered more important, as motivation is key. There's also Italian, which is close to Spanish and isn't difficult to pronounce. It's not really "useful," but fun.

Alternatively, if your daughter has a hobby or interest that may somehow be connected to a foreign language (opera/Italian, for example), I'd focus on that.

Hope this helps, and sorry for the English, I haven't written in Russian in years. :)