Is there an easily accessible text containing a detailed description and derivation of all unitary representations of affine Kac-Moody algebras?



unitary representations of Kac-Moody algebras
Chapter III of the lecture notes on [Kac-Moody and Virasoro algebras](https://arxiv.org/abs/1004.1287) by Antony Wasserman (2010) might well be what you are looking for:



> 
> This course develops the representation theory of affine Kac–Moody
>  algebras and the Virasoro algebra. These infinite–dimensional Lie
>  algebras play an important role in string theory and conformal field
>  theory; they are the Lie algebras of the loop groups and
>  diffeomorphism group of the circle. We adopt a unitary viewpoint and
>  use supersymmetry as the main technique.
> 
> 
> 


