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The real and imaginary parts of the $K_L-K_S$ mixing matrix receive contributions from all three charge-2/3 quarks: up, charm and top. These give both short- and long-distance contributions which are accessible through a combination of perturbative and lattice methods. We will discuss a strategy to compute both the mass difference, $\Delta M_K$ and $\epsilon_K$ to sub-percent accuracy, looking in detail at the contributions from each of the three CKM matrix element products $V_{id}^*V_{is}$ for $i=u, c$ and $t$ as described in Ref. [1]

Calculating the $K_L-K_S$ mass difference and $\epsilon_K$ to sub-percent accuracy