  We present a comprehensive calculation of the $K_L\to\gamma^*\gamma^*$ form
factor in dispersion theory, using input from the leptonic decays
$K_L\to\ell^+\ell^-\gamma$, $K_L\to \ell_1^+\ell_1^-\ell_2^+\ell_2^-$, the
hadronic mode $K_L\to \pi^+\pi^-\gamma$, the normalization
$K_L\to\gamma\gamma$, and the matching to asymptotic constraints. As key result
we obtain an improved determination of the long-distance contribution to
$K_L\to\ell^+\ell^-$, leading to the Standard-Model predictions
$\text{Br}[K_L\to\mu^+\mu^-]=7.44^{+0.41}_{-0.34}\times 10^{-9}$,
$\text{Br}[K_L\to e^+e^-]=8.46(37)\times 10^{-12}$, and more stringent limits
on physics beyond the Standard Model. We provide a detailed breakdown of the
current uncertainty, and delineate how future experiments and the interplay
with lattice QCD could help further improve the precision.
