The fusion category $\mathcal{C}_\ell$ of unitary highest weight projective representations of level $\ell$  of the loop group $LSU(2)$ is equivalent to ${\rm Rep}({\rm SU}_q(2))$ with $q = e^{\frac{i \pi}{\ell + 2}}$ (see [this paper][1], first paragraph p5).   

Now, for any simple object $\rho$ of $\mathcal{C}_\ell$ (characterized by its spin $i \le \ell/2$), there is a Jones-Wassermann subfactor (see this [Jones' survey][2] Section 6, or also [this answer][3]): $$ \rho (L_I)'' \subseteq \rho (L_{I^c})'$$ of index $\frac{sin^{2}(p\pi/m)}{sin^{2}(\pi/m)}$ with $m=\ell + 2$ and $p=2i+1$.  

At spin $1/2$, it is exactly the Jones's original subfactor of index $4cos^2(\frac{\pi}{m})$.


  [1]: https://arxiv.org/pdf/hep-th/9408078.pdf
  [2]: http://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01244-0/S0273-0979-09-01244-0.pdf
  [3]: https://mathoverflow.net/a/144572/34538