The near-infrared transmission spectrum of the warm sub-Neptune exoplanet GJ 1214 b has been observed to be flat and featureless, implying a high metallicity atmosphere with abundant aerosols. Recent JWST MIRI Low Resolution Spectrometer observations of a phase curve of GJ 1214 b showed that its transmission spectrum is flat out into the mid-infrared. In this paper, we use the combined near- and mid-infrared transmission spectrum of GJ 1214 b to constrain its atmospheric composition and aerosol properties. We generate a grid of photochemical haze models using an aerosol microphysics code for a number of background atmospheres spanning metallicities from 100 to 1000× solar, as well as a steam atmosphere scenario. The flatness of the combined data set largely rules out atmospheric metallicities ≤300× solar due to their large corresponding molecular feature amplitudes, preferring values ≥1000× solar and column haze production rates ≥10⁻¹⁰ g cm⁻² s⁻¹. The steam atmosphere scenario with similarly high haze production rates also exhibits sufficiently small molecular features to be consistent with the transmission spectrum. These compositions imply that atmospheric mean molecular weights ≥15 g mol⁻¹ are needed to fit the data. Our results suggest that haze production is highly efficient on GJ 1214 b and could involve non-hydrocarbon, non-nitrogen haze precursors. Further characterization of GJ 1214 b's atmosphere would likely require multiple transits and eclipses using JWST across the near- and mid-infrared, potentially complemented by ground-based high-resolution transmission spectroscopy.