The most general form of Stokes' theorem I know of is proved in the book Partial Differential Equations 1. Foundations and Integral Representations by Friedrich Sauvigny.
The aim in the book is to provide a version of the divergence theorem which holds also in cases where the boundary has certain singularities (as you described: the singular boundary has to have zero capacity). As a precursor they also prove the Stokes' theorem (they credit the proof to E. Heinz!).
Note that this is much more general than manifolds with boundary, it encompasses your cone as well!

The most general form of Stokes' theorem I know of is proved in the book Partial Differential Equations 1. Foundations and Integral Representations by Friedrich Sauvigny.
The aim in the book is to provide a version of the divergence theorem which holds also in cases where the boundary has certain singularities (as you described: the singular boundary has to have zero capacity). As a precursor they also prove the Stokes' theorem (they credit the proof to E. Heinz!).
Note that this is much more general than manifolds with corners, it encompasses your cone as well!