This is what I remember
1. Kagome helps a drunk Sesshomaru home to his appartment.
2. He forces her into a mating.
3. Kagome runs because she isn't willing to give up on her dream.
4. She leaves to become a doctor for hanyo/hanyou.
5. He doesn't know what she looks like, too.
6. For most of the story they aren't together in person. She is learning to be a doctor and he is learning to be a good person.
7. She ends up with Rin. She is a full or Halfling youkai.
8. Inuyasha and Kikyo are now in Sesshomaru's pack.
9. The brothers father is alive, too.
10. All three end up in USA with Kagome and Rin.

I'm pretty sure you want Paper Cranes by midori Haru rated M on fanfiction.net would be MA here.
www.fanfiction.net/s/10253974/1/Paper-Cranes

That's the story. I would have never fond it, without help. Thanks for that help.

You're welcome!