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In this paper we continue the study started in part I (posted). We consider aplanar, bounded, $m$-connected region $\Omega$, and let $\bord\Omega$ be itsboundary. Let $\mathcal{T}$ be a cellular decomposition of$\Omega\cup\bord\Omega$, where each 2-cell is either a triangle or aquadrilateral. From these data and a conductance function we construct acanonical pair $(S,f)$ where $S$ is a special type of a (possibly immersed)genus $(m-1)$ singular flat surface, tiled by rectangles and $f$ is an energypreserving mapping from ${\mathcal T}^{(1)}$ onto $S$. In part I the solutionof a Dirichlet problem defined on ${\mathcal T}^{(0)}$ was utilized, in thispaper we employ the solution of a mixed Dirichlet-Neumann problem.

Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, II: The mixed Dirichlet-Neumann Problem