This thesis presents a first study of the continuum limit of light hadronic physics using a lattice gauge theory simulation with good chiral symmetry. The results are interpreted and extrapolated using both the chiral effective theory and analytic models. Matrix elements of operators of the effective weak Hamiltonian are calculated. The thesis details a combined chiral and continuum extrapolation of two ensemble sets of 2+1 flavour Domain Wall QCD data with inverse lattice spacings around 1.73 and 2.32 GeV. A novel procedure of matching lattice data at unphysical quark masses is used to define the scaling trajectory to the continuum limit. Quantities studied include the pion and kaon masses and decay constants, the average up/down quark mass, the strange quark mass, and the neutral kaon mixing parameter BK. The latter is an important theoretical input to the K band in the unitarity triangle of the CKM matrix. A subset of recent results of ref. [1] in the chiral effective theory needed to perform our fits are re-derived. New methods for the improved determination of the BK matrix element (and other correlation functions), and also for the renormalisation of the relevant four-quark operator for BK are presented.