Here's the thing. I have made approximately 5 versions of Kay. This is the version that I draw currently. If you followed me on dA, you probably saw a glimpse of her before I was banned.
This version doesn't entirely have much of a backstory. Kay was created by Gerald Robotnik as was Shadow the Hedgehog, her "brother" of sorts. I was twelve when I made this story, so give me a break. Around the time that the events of SA2B were occurring, Kay also escaped from the ARK but had nowhere to go. She was separated from Shadow, her only connection to those who were still alive. She goes in search of her long lost brother but doesn't ever actually find him. She instead finds Sin (SiNShadowed's Sin yes), another hedgehog. They become, um... "romantically" involved? ;) And she gives up her search for Shadow when she learns he has amnesia anyways and would have no recollection of her existence. That's where her "backstory" ends essentially. I hate her old backstory, she was way too involved with real characters. Anyways that's Kay for you.