Weakly compact subsets of l-infinity are norm separable

Coordinate evaluations on the l-infinity sequence space form a countable separating family, so a weakly compact set $K$ is metrizable by countable separating family metrizes a weakly compact set. Choose a countable weakly dense subset $D$. Then $K$ lies in the weak closure of $\mathrm{conv}D$, which equals its norm closure by Mazur theorem. Rational convex combinations make that norm closure separable, and every subset of a separable metric space is separable. The weakly dense subset itself need not be norm dense in a nonconvex $K$.