We develop a discrete layer-stripping algorithm for the 2D inverse conductivity problem. Unlike previous algorithms, this algorithm transforms the problem into a time-varying 1D Schrödinger equation inverse scattering problem, discretizes this problem and then solves the discrete problem exactly. This approach has three advantages: (i) the poor conditioning inherent in the problem is concentrated in the solution of a linear integral transform at the beginning of the problem, to which standard regularization techniques may be applied and (ii) feasibility conditions on the transformed data are obtained, satisfaction of which ensures that (iii) the solution of the discrete nonlinear inverse scattering problem is exact and stable. Other contributions include solution of discrete Schrödinger equation inverse potential problems with time-varying potentials by both layer-stripping algorithms and solution of nested systems of equations which amount to a time-varying discrete version of the Gel'fand-Levitan equation. An analytic and numerical example is supplied to demonstrate the operation of the algorithm.