  We are using domain wall fermions to study $K \to \pi \pi$ matrix elements by
measuring $K \to \pi$ and $K \to 0$ matrix elements on the lattice and
employing chiral perturbation theory to relate these to the desired physical
result. The residual chiral symmetry breaking of domain wall fermions with a
finite extent in the fifth dimension impacts these measurements. Using the
Ward-Takahashi identities, we investigate residual chiral symmetry breaking
effects for divergent quantities and study pathologies of the quenched
approximation for small quark mass. We then discuss the $\Delta S = 1$ operator
$O_2$, where chiral symmetry is vital for the subtraction of unphysical
effects.
