The behviors are probably a result of the "new addition" that this girl's father has introduced to the family. She is still dealing with the "loss" of her mother, and another adjustment this soon may be difficult to her.
It seems like your "niece" needs the positive attention of a female role model, a role that you seem a bit hesitant to fill due to the uncertainty of the length of her time with you. To make the situation even more difficult, it is uncertain whether or not your niece will be around much if they decide to move, and, it seems what you are saying is that you don't want her to have to deal with losing another female in her life if she becomes too close to you?
Kids need someone to show them that they care. Since she does not have her mother around and her father is working and dating someone else, the only stable female role model that she has in her life is you, as her "aunt". You won't replace her "mom", but you can provide her with the stability, structure, and love that she seems to need in her life while she is with you. It may not seem like much, but just by being there and showing that you care makes the difference. You and your family are what this little girl has to help her through this difficult transitions in her life, so being there for her, supporting her, and being the structure that she needs is the best thing that you can do to help the situation. It may seem trying with the crying and the whining, but it seems like she is able to process her feelings with you . . . so provide that listening ear, and let her father know what's going on as well. The trip to the doctor's is a step in the right direction, but this little girl is dealing with a lot right now . . .so maybe some kind of counseling may be in the future as well.