Why don't you just step in?
Sure, a kid can attempt to handle it themselves... but at a certain point, per the parent gauging it.... you need to KNOW when to step in.
This friend of your daughter's..... is very OBSESSIVE.
Your daughter already tries to handle it.
What more, do you want her to do???? To handle it on her own???
The thing is: That other girl, will NEVER NEVER NEVER be satisfied with whatever your daughter does, or does for her, nor no matter how she proves she cares... because this other girl, is being very controlling and obsessive about BULLYING your daughter.
And there comes a point, where you need to step in.
I have a daughter who is 9. She is very good about choosing friends and speaking up. She is very mature.
BUT... there was a girl that was always PUSHY and controlling and bossy and just would not leave her and her other friend, alone.
Even the Teacher, would get irritated about this girl, because the girl was just really egocentric.
So... one day, I was there at school.... I saw the girls telling her to stop it etc. And she would not. And this girl was causing a CONTINUAL stress to the other girls. Everyday. Even if they tried to avoid her in various manners... the other girl would not stop.
So as I said, one day I was at the school and saw this tit-for-tat going on that the other girl was doing. SO... I went up to the girl, (all the kids know me), and I spoke to her, with ALL the girls around me. I told the girl "You cannot force, other people to do what you want, you cannot force other people to play with you if they do not want to, if they tell you to stop- then stop. They are TRYING to be polite, but you do not allow this. This is life. When you are an adult, or working in an office, you cannot do things like this. EACH person, has a right... to say, "no." And you need to accept that." Then I told the other girls, that they can say no, or stop it, to her. And its okay, they need to speak up to those who are bossying them. My daughter and all the other girls, TRIED so many times... to be polite to the other girl and to try various things, to avoid her. But this other girl, is sooooo bossy, and to so many other kids. So I stepped in. And then, it stopped. That other girl, was causing SOOO much STRESS to the other girls, my daughter included... that it was just, a totally burnt out situation already. And I explained to the Teacher. Too. It was fine.
Now, your daughter's "friend" is FORCING your daughter, to PROVE to her, how much she loves her like a sister and how much she cares. That is... really, pushy. And, your daughter should NOT be someone else's door-mat. Look at all the hoops and stress, your daughter has to go through... JUST to satisfy, that other girl? That is not, a "friend." That is emotional manipulation that is dysfunctional.
Egad. If that were happening to my daughter... I would step in and speak to that other girl... as mannered as I can, but getting the point across.
When I was a Teen... there was a boy that was acting like that to me. After trying to handle it myself for so long, I told my Dad. My Dad... stepped in and talked to that boy. It stopped. Sure, this is a boy... but think about it another way... would you want your Daughter... to be like this with a boy????? What if a boy were treating her that way???? What would you do???
As you see, this same situation... could be with this girl or a boy. So then, how would you expect your daughter to handle it?
What more, do you want your daughter to do, to handle it??? On her own?
Good grief... if another child were doing that to my daughter... I would be upset. It is wrong. And I would step in.
And I would not expect... my daughter, to be a "social worker" to that other girl. A child also needs to know... that they can "choose" friends, and be wise about it too.
My kids are 5 and 9.
I have always taught them... how to choose friends, how to say no, and that they do not have to be someone else's door-mat. If something is amiss, they need to tell me or the Teacher.
KEEP that e-mail/text from that other girl & your daughter's response. Because, then it is "proof" of her behavior toward your daughter... and just in case, she starts to lie, to get your daughter in trouble.
That other girl, is an emotional BULLY to your daughter.
Get her away!
Step in.
Your daughter, is suffering.
Put it this way... not even ADULTS, may be able to handle a "friend" or co-worker like that, nor know how to.
So, why should an 11 year old... be on their own, to handle this? That other girl, is TOTALLY dysfunctional... and is using your daughter as her DOOR-MAT.
End it.
And it seems your daughter is just getting HARASSED!
And bullied.
Bad, combination.
Don't let your daughter be a victim. Nor a door-mat.
And teach her and guide her.