The Drinfel'd centres of String 2-groups

Let $G$ be a compact connected Lie group and $k \in H^4(BG,\mathbb{Z})$ a cohomology class. The String 2-group $G_k$ is the central extension of $G$ by the 2-group $[\ast/U(1)]$ classified by $k$. It has a close relationship to the level $k$ extension of the loop group $LG$. We compute the Drinfel'd centre of $G_k$ as a smooth 2-group. When $G$ is semisimple, we prove that the Drinfel'd centre is equal to the invertible part of the category of positive energy representations of $LG$ at level $k$ (as long as we exclude factors of $E_8$ at level 2).