My first thought was: this little girl is a BRAT!!!! As the nanny it's hard, but you can stop the behavior.
Stick to your guns, if she's whining at you, tell her you don't understand her when she's whining, then leave the room. If she continues to whine at you, put her in her room and tell her she can come out If, and only if, she stops whining. If she's having a fit about tying her shoes give her a time limit. She has 5 minutes to put her shoes on or you're not going to____________. If she doesn't comply, it's her loss.
When at the park and she whines that nobody will play with her tell her that nobody likes a cheater. If she whines as you drive by a park tell her that only good girls that don't whine get to go there. If she starts whining about being "hurt" then have her sit quietly next to you for a while. If she starts wanting to play again because the other kids are playing and having fun, tell her nope, you're hurt, you have to sit and watch. Also I'd tell her that if she starts whining about anything while at the part you will leave immediately and she can spend time in her room.
I'd even consider talking with her parents and see if you all can work on her whining. Most likely she's whining at them too. Y'all need to be a united front and shut this down now. I don't have a specific link, but Super Nanny should have some great tips on how to stop this problem behavior.
BTW, this 8 year old little girl sounds like my 4 1/2 year old son. We've done all my suggestions with him and one or another has worked with him just about every time.
Hope this helps,