  The $e^+e^-\to K^+K^-$ cross section and charged-kaon electromagnetic form
factor are measured in the $e^+e^-$ center-of-mass energy range ($E$) from 2.6
to 8.0 GeV using the initial-state radiation technique with an undetected
photon. The study is performed using 469 fb$^{-1}$ of data collected with the
BABAR detector at the PEP-II $e^+e^-$ collider at center-of-mass energies near
10.6 GeV. The form factor is found to decrease with energy faster than $1/E^2$,
and approaches the asymptotic QCD prediction. Production of the $K^+K^-$ final
state through the $J/\psi$ and $\psi(2S)$ intermediate states is observed. The
results for the kaon form factor are used together with data from other
experiments to perform a model-independent determination of the relative phases
between single-photon and strong amplitudes in $J/\psi$ and $\psi(2S)\to
K^+K^-$ decays. The values of the branching fractions measured in the reaction
$e^+e^- \to K^+K^-$ are shifted relative to their true values due to
interference between resonant and nonresonant amplitudes. The values of these
shifts are determined to be about $\pm5\%$ for the $J/\psi$ meson and $\pm15\%$
for the $\psi(2S)$ meson.
