This reminds me of a story--
I worked with a woman, Lucy, who was married with no children. She and her husband had a nicely decorated house with very nice furnishings.
One evening they invited a co-worker of her husband's and his wife and 8-year-old daughter to dinner. The parents had no control over the daughter. After dinner, while the grown-ups were having coffee in the living room, the kid started running through the house in a circle from the living room to the dining room to the kitchen to the entry hall and back into the living room.
The parents ignored the weak "oh, be careful honey, I don't want you to fall and hurt yourself" remarks that Lucy made. So she got up and went into the kitchen apparently on some errand and layed in wait until the kid came charging through again. She grabbed her by the arm and pulled her right up close so she was hissing directly into her face and said something like, "Listn you little brat, you know I don't want you running in my house. You go right in there and sit down or you're going to be sorry. Do you understand?" Then Lucy got whatever she came into the kitchen to get and when she got back to the living room the kid was seated on the couch next to her oblivious parents.
The kid probably told the parents what happened on the way home, but Lucy didn't care because
1) the problem of running through the house was solved as far as Lucy was concerned
2) Lucy didn't care if she ever saw those people again
3) the parents needed to know that not everyone found their daughter as charming as they did
4) maybe the kid learned from it and became civilized
Back to the point of the thread (finally). Get the kid and the kid's mother before the chasing starts and explain in no uncertain terms that it is not allowed and what will happen if it does. If the chasing continues I would charge into the yard at a run and snatch that kid up and march her into the house and raise hell with her and her mother.
If you act like a namby-pamby old lady, that's how you will be treated. If you act like those fowl are your property that must be treated properly out of respect for you, that's how they will be treated.
After you scare the kid half to death, let some time go by and, if they ever visit you again, offer to share some simple chores. This will let the kid know that these are still your birds and she only approaches them if you say she can and, just as importantly, that you love your niece and want her to share the fun of seeing the animals up close and enjoying them with you.