No matter what the case may be with that older child, you are not a 'social worker' nor trained to be one... and it is not your 'obligation' to mind another child, all the time... if you do not feel it is appropriate or safe or healthy for your own, child.
Your child comes first.
And your sense of what is appropriate or not.
And what your sense of boundaries are.
AND... the PRIMARY thing here... is your own child, does not seem 'happy' either, with this situation.
As you said, even your child gets fed UP, with this older child.
It seems, both you and your almost-3 year old daughter, is this 11 year olds, "babysitter."
And social worker.
And chaperone.
And, play mate
And... whatever else the girl is expecting from you/your child, as she constantly sits and waits for you at your home.
You also have no privacy, nor any sense of peace.... because you KNOW she will, unannounced, pop-up at your house.
Anytime.
Again, not everyone, can or has to... take in neighbor's kids, especially if they don't feel it is appropriate.
IF your 'Mommy radar' is tweaked... then listen to that.
If it were me: I would, get fed-up too, because it means, having to REVOLVE my entire household and child's schedule or my schedule... around this 11 year old visitor... unannounced. AND having to constantly keep constant supervising over that child while with my own child... because, you cannot trust that older child.
So, if it impedes my life/my child's life that much and CONSTANTLY.... I would simply tell her she cannot visit or must go home... because "we are busy..."
How long, how many hours is this older child at your house??? And your having to CONSTANTLY watch over her while with your child???? I mean, isn't that really confining... and in your own house???
An 8 year age difference.
Is a lot.
I would just tell the girl... "sorry, we are busy" or "you need to go home." or "My daughter is too young, to play with you."
We had a neighbor's kid, that would actually just watch our house, and as soon as we were home, would come running up to our garage even before we got into our front door ourselves, to "play" with my kids.
Okay fine.
In the beginning it may be fun and a novelty for my kids and that kid.
But after awhile, it is, really not fun.
Even my kids, would dread... going out the front door or in plain view of that kid's house, because they did not want to be pounced upon. And sometimes they did not want, visitors or anything either.. nor have to entertain another child.
Sure I spoke to the Mom. But I don't think she realized.
We would tell the child we are busy.
Kids don't hear what they don't want to hear.