I think what we all need to learn is that we can't fix or change anything. Your daughter has attached ,for now, to your mother, and that is a better alternative than living on the streets.
I truly understand the pain, of being rejected and painted black by your daughter. Bpds can be very convincing, my BPD convinced my mother that I was holding her back from her life. My d22 moved out with her homeless boyfriend, and is now on welfare, and my mother seems to understand that everything wasn't as my d22 portrayed it.
Look up information on this website on Wise Mind. If you can treat your d20 like an acquaintance, it may relieve some of the tension. You wouldn't start accusing an acquaintance of not living a lifestyle you approve of, or point out when they lie, or how come they didn't remember their mom's birthday. You might THINK those things, but you would wave your hand and say, oh well, that is THEIR life.
I have, after almost 4 months, started communicating a little better with my d22. She has come back home a few times, shared meals with us, we still refer to her bedroom as hers, her photos are on the wall. But I am trying VERY hard not to change her, although I have offered her help in some aspects of getting her life back together.
And birthdays are so important in my family. My d22 was living on the streets for hers, and for Christmas. Yeah, it is empty
Hang in there, stay cool
. She hasn't forgotten you, she is taking a break, and she is with family.