  We use HPQCD's recent lattice QCD determination of $B \to K$ scalar, vector
and tensor form factors to determine Standard Model differential branching
fractions for $B \to K \ell^+\ell^-$, $B\to K \ell_1^+\ell_2^-$ and $B \to K\nu
\overline{\nu}$. These form factors are calculated across the full $q^2$ range
of the decay and have smaller uncertainties than previous work, particularly at
low $q^2$. For $B \to K \ell^+ \ell^-$ we find the Standard Model branching
fraction in the $q^2$ region below the squared $J/\psi$ mass to exceed the LHCb
results, with tensions as high as $4.2\sigma$ for $B^+\to K^+\mu^+\mu^-$. For
the high $q^2$ region we see $2.7\sigma$ tensions. The tensions are much
reduced by applying shifts to Wilson coefficients $C_9$ and $C_{10}$ in the
effective weak Hamiltonian, moving them away from their Standard Model values
consistent with those indicated by other $B$ phenomenology. We also update
results for lepton-flavour ratios $R^{\mu}_e$ and $R^{\tau}_{\mu}$ and the
`flat term', $F_H^{\ell}$ in the differential branching fraction for
$\ell\in\{e,\mu,\tau\}$. Our results for the form-factor-dependent
contributions needed for searches for lepton-flavour-violating decays $B\to
K\ell^-_1\ell^+_2$ achieve uncertainties of 7%. We also compute the branching
fraction $\mathcal{B}(B\to K\nu\bar{\nu})$ with an uncertainty below 10%, for
comparison with future experimental results.
