In a previous paper (Paper I) we presented a perturbative analysis of the collapse of a molecular cloud-core threaded by an ordered magnetic field, obtaining a semianalytical solution applicable over a moderate range of temporal and spatial scales. In the present paper we supplement this analysis with a numerical solution of the magnetohydrodynamic (MHD) equations that include the effects of ambipolar diffusion, valid in the region where magnetic effects dominate the dynamics of the collapse. We focus on the formation of a flattened disequilibrium structure (''pseudodisk'') around the central protostar. The numerical solution gives dimensionless values for the radius of the pseudodisk as a function of time. Combined with the analytical scaling laws found in Paper I, these results provide in the small time limit a simple power-law expression for the dimensional radius of the pseudodisk as a function of the initial magnetic field B0 and effective sound speed a of the unstable molecular cloud core. We tabulate in nondimensional form the velocity, density, and magnetic fields as functions of the radius, polar angle, and time for two values (chi = 11.3 and infinity) of the ion-neutral coupling constant. We apply the results to the density and magnetic field structures on the astronomically interesting scale of a few hundred to a few thousand AU around protostars with mass in the range 0.57-2.0 M.. The resultant magnetic field topology causes us to speculate on the importance of neutral-ion slip, ohmic dissipation, and reconnection in the overall problems of the loss of flux and the isolation of the magnetic fields in the pseudodisk (and smaller centrifugal disk) from their interstellar origins. We conclude by comparing our results with observations of flattened dense structures around young stellar objects in various stages of evolution.