Wow, that's awful, there's no point spending ten years raising your child if you are going to end up with no good relationship with them.
As a single parent, I do perhaps understand that a decade of parenting solo is tough, and perhaps, she feels so exhausted from the relentless responsibility that she is not thinking long term. It's a shame, I know I tried to date when my dc were between 8 and 16 and it was just a disaster. If I was even on the phone to a guy I met from work, my DD would start crying, telling me I was ignoring her. I gave up totally though. I thought, right, this isn't worth it. I have a good relationship with her now that she's an adult but I think it can be tough, when you feel you've been parenting for a decade and there's still a decade to go.
Could you and her sister support each other in a plan to take the 10 year old every second weekend (each, so, once a month for you, once a month for the sister) and that way she has planned slots free to have a bit of an adult life.
I have no ''adult life'' and I'm ok. But there were times where that felt hard I know.
Also, part of the problem can be that all socialising in your circle can be couples so sometimes, a man, any man can be a ticket to being included suddenly.
It's shit.