I think you have had many good responses here and am glad you have decided to reconsider. I wanted to also add my tuppence worth.. feel free to ignore..
Therapy is not just for when you are struggling, it is a time and space to reflect and many people believe we would all benefit from this all the time. I think you would benefit from time and space to reflect and consider the possibilities for your family. Ideally, family therapy, even if you are the only one attending.
Home: her home is not England, growing up as an expat kid is not the same as growing up in the place where all your extended family live, where you were born or the country who's passport you hold. Her experience of belonging, home and family are very different to yours. Try to see this from her perspective and let go of the idea that going back to England will reset all of this in her... recognise the person she is, the one that her life experiences to date has moulded.
Boarding school: full on scheduled time with short time slots to fit in family visits / exeats is nothing like growing up around family. Even if she has good relationships with some family members, this is way to build polite interactions, not intimacy and deep relationships. This will not fill her emotional needs. Do not underestimate the link between boarding school and attachment issues, anorexia, mental health issues.
The danger of roles: You are all playing roles, you describe yourself, your estranged daughter, your ex, his girlfriend, the daughter (s?.. not sure if there is more than one with you) .. these roles are not fixed, framing everyone in this way is harmful, it affects their ability to step outsides frames, for you to see them in different ways and it affects how other people behave.. your younger daughter who you describe as your shadow may be your shadow in response to the relationship you have with your older daughter.. it may not be her true self, whatever that may be.. but the situation will be impacting her too. You say your older daughter, is like her dad and you do not like her dad... ergo... you do not like her... you may not say this in words, but if you think this, she will feel it. She has picked a side, but she shouldn't have to.
She is 11!! She needs you to guide her, to support her, to teach her.. she didn't ask to be born, to be moved around, to have fleeting relationships with relatives she is expected to identify with, to have her parents marriage and therefore her stability, her identity and understanding of herself break up. She has not asked to be sent away from her parents, sister(s) and friends.
Read, read, read.... How to talk so your kids will listen, Hold on to your kids, there are probably many many more... even when they are adults, we can support, listen and be there for our children... allowing them to be independent is not the same are teaching them to develop healthy relationships.
I wish you the very best... no person, no family, no situation is perfect. I am struggling on a number of fronts and finding it hard to let go of mistakes, decisions that had unintented consequences and to see the choices that are in front of us rather than the paths that are laid out from our previous perspectives.. we are not where we thought we would be a few years ago so the things we thought then are no longer 100% correct or even relevant. I hope you can reframe your situation and go forward positively doing best for your family.