  We derive a set of light-cone sum rules relating the $S$-wave $B\to K \pi$
hadronic form factors to the $B$-meson light-cone distribution amplitudes
(LCDAs), taking into account the complete set of LCDAs up to and including
twist four. These results complement the sum rules for the $P$-wave $B\to K
\pi$ form factors obtained earlier. We then use the new sum rules to estimate
the $S$-wave contributions to $B\to K \pi\ell\ell$ decays as a function of the
$K\pi$ invariant mass. We pay particular attention to the fact that the
$S$-wave $K\pi$ spectrum cannot be modelled by a sum of Breit-Wigner
resonances, and employ a more consistent dispersive coupled-channel approach.
We compare our predictions for branching ratios and angular observables with
LHCb measurements in two different kinematic regions, around $K^*(892)$ and
$K^*_0(1430)$. We observe an overall compatibility and discuss possible
improvements of our model to obtain a better description of the $B\to K\pi$
form factors over a large kinematic range.
