Katherine was taken as well when her mother, shortly after surrendering Lucy, was deemed ill fit to take care of her. Her mother eventually got things together, yet didn't tell Kath that it would be unlikely the her sister would return as well due to surrendering her. Katherine is unaware of what exactly happened at the orphanage, she only knows that there was a particularly brutal fight amongst some of the children. She and her mother did try to get her sister back on some occasions, though couldn't find her after she was dispersed into the system following the incident.

She is somewhat aware that her father has another child: Lumina would pop up due to a shared surname and parent, though likely before she is put into the system. It also acts as the only window Kath will ever have into her father's life (unless more pink-haired relatives pop up somehow.) after leaving. Alternatively we could just have Kath completely unaware of Lumi's existence and have Lucy telling her be the way she learns of her.

I cannot wait for the sheer mindfuck their family tree is gonna become once Lucy and Kasen get their horns back, yet find their younger sister doesn't have them. The whole "Secret Oni" revelation is going to rock the foundations of how Katherine views herself.

This is going to be fun.