It has long been shown that the extreme ultraviolet spectrum of the ionizing stars of HII regions can be estimated by comparing the observed line emission to detailed models. In the Galactic Center (GC), however, previous observations have shown that the ionizing spectral energy distribution (SED) of the local photon field is strange, producing both very low excitation ionized gas (indicative of ionization by late O stars) and also widespread diffuse emission from atoms too highly ionized to be found in normal HII regions. This paper describes the analysis of all GC spectra taken by Spitzer's Infrared Spectrograph and downloaded from the Spitzer Heritage Archive. In it, HII region densities and abundances are described, and serendipitously discovered candidate planetary nebulae, compact shocks, and candidate young stellar objects are tabulated. Models were computed with Cloudy, using SEDs from Starburst99 plus additional X-rays, and compared to the observed mid-infrared forbidden and recombination lines. The ages inferred from the model fits do not agree with recent proposed star formation sequences (star formation in the GC occurring along streams of gas with density enhancements caused by close encounters with the black hole, Sgr A*), with Sgr B1, Sgr C, and the Arches Cluster being all about the same age, around 4.5Myr old, with similar X-ray requirements. The fits for the Quintuplet Cluster appear to give a younger age, but that could be caused by higher-energy photons from shocks from stellar winds or from a supernova.