The detailed knowledge of plasma heating and acceleration region properties presents a major observational challenge in solar flare physics. Using the Ramaty High Energy Solar Spectroscopic Imager (RHESSI), the high temperature differential emission measure, DEM(T), and the energy-dependent spatial structure of solar flare coronal sources are studied quantitatively. The altitude of the coronal X-ray source is observed to increase approximately linearly with a gradient of +0.2 arcsecond/keV over an energy range from 10 to 25 keV. It is shown that despite an isothermal model providing an adequate fit to the thermal X-ray spectrum observed by RHESSI, such a model cannot account for the observed imaging data, hence requiring non-isothermal coronal sources where the temperature increases with altitude. We show for the first time that in order to correctly determine DEM(T), imaging information must be used. A thermal bremsstrahlung X-ray emission model with inhomogeneous temperature and density structures is developed to simultaneously reproduce: i) DEM(T), ii) height above the limb as a function of energy, and iii) vertical extent of the coronal source versus energy. We find that the temperature-altitude gradient in the region is +0.06 keV/arcsecond (~1 MK/Mm) and the number density falls approximately an order of magnitude. Similar height-energy trends in other flares suggest that the majority of coronal X-ray sources are non-isothermal and have strong vertical temperature and density gradients with a broad DEM(T).