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

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Chapter III of the lecture notes on Kac-Moody and Virasoro algebras 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.

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