The Khmaladze Transformation provides a general method for transforming parametric empirical process into an asymptotically distribution-free process. Distribution-free goodness-of-fit statistics can then be obtained from this transformed process, i.e. their limit distribution does not depend on the parametric family under consideration. However, this transformation involves matrix inversion and if degeneracy is present it seems at first that the transformation is no longer well defined. The Laplace family of distributions (i.e. symmetric and asymmetric Laplace) are a practically important family where this apparent complication arises. In the context of the Laplace family, we demonstrate that even though this component of the transformation appears problematic, the transformation is still well defined. We show explicit form of this transformation for these families and show with simulations that goodness-of-fit statistics obtained, demonstrate desired distribution freeness.