@littlek,
In this case, I apologize if I upset you, but I must warn you from personal experience: If you let things like this slide, they can get very grim indeed. My mother, for example, used to own a business together with her two brothers. There were always conflicts of interest -- normal stuff for a business, but near impossible for the family to handle as a family affair. My mother
always ended up sacrificing her legitimate business interests for the sake of the family, whom she adored. (You kind of remind me of her in this regard, except perhaps for the adoring part.) Her reward was that she lost it all: The business went bankrupt; my grandmother, the mother of the three, got extremely depressed over this and died; my mother and her brothers have barely been on speaking terms since the bankruptcy nine years ago.
[patronizing big brother rant]
This is the last thing you want -- for yourself or your family. I couldn't for the life of me think of any business interest that's worth this kind of distress. So, with my own family history in mind, I recommend with the utmost severity that you take one of the following two options.
The best option is that you try to end the nannying business with your sister as soon and as permanently as you can.
If you can't do that -- if, for example, you need a fallback in case something goes wrong with your teaching career -- the second best option is this: Draft a written contract (your library will have legal form books for this purpose), get your sister to sign it, and stick to it religiously. That way, at least both parties have a written record of their rights and obligations, and no longer have to "feel" what they are.
[/patronizing big brother rant]
I know all these things are much easier for me to say than for you to do. I wish you the very best!