Kids, don't make plans.
The Moms, do.
ie: when the girl says that she rather go back to her house and take your daughter with her, you tell her no, the play date is at your house.
But if she wants to leave your house, fine, You will take her back home, but your daughter is staying home. And that's that.
And, if there is any change of plans, both Moms have to agree first. Since the play date is at your house, the other girl's Mom, has her own plans now, too.
On the few occasions when that has happened at our house, (my daughter has a friend that gets bored easily), I tell her "No, your Mom has plans now, you are here, and she needs to get things done..." and I call that other Mom, and I tell her her daughter is "bored" and says she wants to play date back at her house. The other Mom then says "That's my daughter! Bored. Too bad. Tell her no, I am busy. I will pick her up at the planned time."
And I tell the girl "have fun while you are here. Its not like you can come here everyday. So be positive."
How old are your daughter and her friend?
My daughter is 10.
They can entertain themselves. At least my daughter is very good at that. And when her boring friend comes over, well, I do not cater to it.
My daughter and she, will sometimes just hang out. I may make suggestions, but the girl is such the picky person. So oh well. So then I tell her "Well, you both can just hang out and stare into space. You know what we have here, you know what you can do, and its your choice."
But interestingly, both my daughter and she do, have fun. They just like each others company. And the other girl, likes, to come here. Despite her OWN tendency to get bored. That is her problem. But I will not make it into my problem.
And at least my daughter, does not get phased by it or her friend. My daughter laughs about it, says she knows her friend is like that. But they just like to hang out and have each others company. The girl is nice. She just is not good at entertaining herself or using her imagination.
For you, well you can let the girl leave your house when she is bored, take her home.
BUT, your daughter stays home.
There will always be other homes or families, that have more or less than you, and other siblings or not, or other people's homes are neater or bigger or smaller or not. So what.
Friendship is not based on those things.
And having fun, is not based on that either.