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CT missed the most important point here. The problem is that you and your husband are not acting together on this, for whatever reason. If you help this girl, it will happen because you do it together. Right now it sounds as if the two of you are not communicating at all and have different strategies about her. It's time to change that.
The two of you have to have a long, serious discussion about the niece, without her around to interfere. You have to make this happen. Your husband will not. He is not thinking, just feeling, so you will have to do the thinking for him. Hopefully, he's aware enough to recognize truth when he hears it. Tell him that you know that he loves his niece, that that makes you happy, and that you want to love and nurture her, too. Everybody agrees that we want what it best for her.
Then you have to get him to see that what is best for her may not be what she thinks she wants. She is young and traumatized, and she is trying to protect herself by recreating a dominant male who is focused on her. This is all she knows. You have to show your husband that this will hurt her in the long run. She should learn that love is inclusive, not isolating. He must help you stop her competition for attention. He must ignore her when she tries to distract him from you. He has to remember that you are his wife, and then you both have to be prepared for the acting-out that will ensue when she feels threatened by her demotion. In the movies, this would be resolved by your taking her shopping.
Finally, both you and your husband should remember that 13 doesn't last forever. You can try to be compassionate to both of them, and he can try to be more firm. Tell each other that you know the next year will be a difficult test, and that you want to come out the other side a stronger, more loving couple.
Soon, she'll have a boyfriend, and you'll have a whole new set of problems.