Do not expect alignment with current American leftist mores here. Gin has already called her a faggot and said women can get pregnant, and she can't. (This instigates her comment about Mary.)
(Reminder that actually poor and oppressed people are not politically correct.)
She didn't know her mother, and was brought up in the foster system. She wants to know why. For Kiyoko's sake, and her own too, she wants to find the mother.
They have only one clue: a locker key that was wrapped up in the blanket with Kiyoko.
They find a travel bag with some sexy clothing, a few pictures, and some business cards for a hostess club.
With that clue, they take a train in that direction. But when the trains are stopped by snow, Miyuki is spotted (by someone from her old life?) and they abandon the train.
But Gin knows the groom. He's the one who arranged his fall from grace.
Gin goes to kill him with a bottle, but is saved from his own rage when one of the caterers pulls a fun and tries to shoot Ohta, and the groom jumps in the way of the bullets and gets himself killed.
Miyuki had adopted a stray cat. Something went wrong; maybe her father got rid of the cat, or maybe something else happened, but she ended up stabbing her father.
She ran away after that and has been on the lam ever since.
The nurse makes small talk with Miyuki about Kiyoko. Her name is Kiyoko too.
When Gin goes to pay Hana's bill he crosses paths with the nurse, who looks like she's just seen a ghost.
It's his daughter.