X-ray and Extreme Ultraviolet emission spectra resulting from energetic sulfur and oxygen ions precipitating into the Jovian atmosphere are calculated. Monte Carlo simulations of the energy and charge relaxation of downward ion fluxes are carried out, using updated collision cross sections for stripping, electron capture, and target ionization. Energy and charge distributions of precipitating sulfur ions are presented for the first time and the equilibrium charge model is shown to be inadequate. X-ray emission spectra are calculated for different sulfur and oxygen mixtures and for different initial entry energies. Satisfactory agreement with both Chandra and XMM-Newton observations is obtained by an equal population of sulfur and oxygen ions with energies between 1 and 2 MeV/amu. The agreement provides a reconciliation of the two spectral data sets and the inferred initial energies are consistent with the view that the ions are magnetospheric in origin and have been accelerated to MeV/amu energies.