I may draw some criticism for what I am about to say, so I have to give a little background: I am a retired US Air Force Master Sergeant who believes that respect cannot be demanded - even by one’s superiors - but that courtesy can.
I always told my new troops that I wanted them to respect me and that I would strive to earn their respect; however, whether or not they respected me, they would behave toward me with the courtesy and deference befitting my station as a senior NCO and their supervisor. I never had any trouble with any of my troops after telling them that, and they were courteous to me; similarly, I was always courteous to them because I had to set the appropriate example.
Now, regarding the young girl at hand: This child sounds as though she is rapidly becoming sociopathic, if not psychopathic, and she needs counseling and therapy immediately. You, however, might consider taking her aside and speaking privately to her in this vein:
"Jane (or whatever her name is), what I have to say to you is just between us; your Mom and Dad don’t have to know about this because it is private.
"It seems to me that you do not like adults. I understand that; I used to hate the fact that I had to do everything my parents and teachers told me to do because they were the adults and I was just a stupid kid. I am not asking you to love me or even like me. I want you to respect me, but I can’t make you do that - I have to earn your respect, and I will do whatever I have to do to earn it.
"The first thing I’m going to do is to promise that I will always treat you with the same courtesy I show any other adult. When I speak to you, I will call you “ma’am” just as I would when speaking to any other woman. I will always assume that you are a lady until you prove to me that you are not, and I will ignore any insult unless you make it very clear to me that you intended to insult me. When that happens, I will do the same thing to you that I would do to any other adult who purposely insults me.
"In return, I expect that you will show me the same courtesy that I show you - nothing more, nothing less.
"Now: if that sounds like a good deal to you, say so. If you turn around and walk out without saying anything at all, I’m going to assume that you think it’s a good deal and you will hold up your end of the agreement.
“If it doesn’t sound like a good deal to you, say so; I won’t expect you to show me any courtesy, but I will continue to behave toward you with the courtesy you deserve because I have given my word that I would, whether you do or not.”
Then you must do the most difficult thing possible: you must keep your word. Even if the little snot never gets over her sociopathy, it will not be because she did not have a good example to follow - you.
If she does insult or offend you, and makes it clear that she intended to, do whatever it is you would do if another adult insulted you.
(During my career, I backed someone who was discourteous to me up against a wall and gave him a tongue-lashing that left him gasping for breath. I don’t usually have to result to violence, and I don’t advocate using it against children, but you do have to get her attention; if that means getting in her face, just try not to hurt her.)
You may have to explain to her parents that you are trying to teach their daughter that there are people in the world who will not take kindly to her behavior, and that it is better that she learn the lesson from a close family member who will not harm her, rather than from a stranger who might feel constrained to hospitalize her. You need to stand up for yourself, because you are deserving of courtesy and because your niece must be taught; otherwise, she will run into a very sharp learning curve later in life - one she may not survive.
Privately, pray that God would show His love for her through you. Pray it every day, fervently, sincerely; it does work. I know, because it works for me every time I do it.
Good luck; you’re going to need it!