One of the outstanding problems in modern nuclear physics is to determine the properties of nuclei from the fundamental theory of the strong force, quantum chromodynamics (QCD). Skyrmions offer a novel approach to this problem by considering nuclei as solitons of a low energy effective field theory obtained from QCD. Unfortunately, the standard theory of Skyrmions has been plagued by two significant problems: (1) It yields nuclear binding energies that are an order of magnitude larger than experimental nuclear data, and (2) it predicts intrinsic shapes for nuclei that fail to match the clustering structure of light nuclei. Here we show that extending the standard theory of Skyrmions, by including the next lightest subatomic meson particles traditionally neglected, dramatically improves both of these aspects. We find Skyrmion clustering that now agrees with the expected structure of light nuclei, with binding energies that are much closer to nuclear data.