There's nothing complicated to the phrase "to do anything with". It means what it says. You could substitute "accomplish" or "achieve" for "do".
In the example you give, Annie Sullivan is saying that teaching Helen Keller is going to be very difficult. Your idea about manners could be inferred from the rest of the letter, but has nothing to do with the sentence you quoted.
My car is badly wrecked. I have it towed to a repair shop. The mechanic says, "I don't think we'll be able to do anything with it."
I'm unsatisfied with my haircut. So I go to a different barber and say "Can you do anything with my hair?"
I'm interviewing a chef, and ask "Can you do anything with chicken that's different and exciting?"