I believe it is or was on spark but i cant find it.

They go to a school, not sure if it is like high school or a university, in this school each student is set up with another student. These pairs eventually become mates. Kagome is paired with Sesshomaru. At first they hate each other. They end up living in a dorm together inside the school. I believe it has around 25 chapters not sure though. I also keep thinking that the title has shikon in it but again im not sure. Ive searched spark under shikon but none of the fics coming up sound familiar.

It might possible be :

Shikon Academy » by POCKYLUVR101 reviews
Kagome is a new student at Shikon Acd. There she meets people of all kinds and learns that she's a miko. Of course she has friends that mean a lot to her. But as stuff is thrown her way she realizes that she needs them now more than ever. Please read.

I also remember one that is similar:

Breeding School…
They all go to a special school, where secrets are held, and magical things take place. The older students say nothing, so they are all left in the dark, but there is something strange about this school - Breeding School High… SessxKag
Inuyasha - Rated: M - English - Romance - Chapters: 26 - Words: 159,809 - Reviews: 1066 - Updated: 11-17-08 - Published: 2-22-06 - Sesshomaru & Kagome

There are several others, but they are escaping my memory at the moment.

It looks more like the second story's plotline. A very good read. Too bad it was never finished.

It is breeding school thank you. And it is a shame that it has never been finished

Oh my gosh, it's been so long since I've read Breeding School! It still makes me sad that the story was never finished.