We describe \emph{a posteriori} error analysis for a discontinuous Galerkinmethod for a fourth-order elliptic interface problemarising from a linearized model of thin sheet folding.The primary contribution is a local efficiency bound for an estimator that measuresthe extent to which the interface conditions along the fold are satisfied,obtained via a new edge bubble function that is only globally $H^1$ and locally $H^2$,in contrast to existing constructions requiring global $H^2$ regularity.We subsequently perform a \emph{medius} analysis to derive improved \emph{a priori} error bound under minimal regularity assumptions on the exact solution.The performance of the method is illustrated by numerical experiments.