We use the best available X-ray data from the intermediate polar EX Hydrae to study the cooling-flow model often applied to interpret the X-ray spectra of these accreting magnetic white dwarf binaries. First, we resolve a long-standing discrepancy between the X-ray and optical determinations of the mass of the white dwarf in EX Hya by applying new models of the inner disk truncation radius. Our fits to the X-ray spectrum now agree with the white dwarf mass of 0.79 M$_{odot}$sun determined using dynamical methods through spectroscopic observations of the secondary. We use a simple isobaric cooling flow model to derive the emission line fluxes, emission measure distribution, and H-like to He-like line ratios for comparison with the 496 ks Chandra High Energy Transmission Grating observation of EX Hydrae. We find that the H/He ratios are not well reproduced by this simple isobaric cooling flow model and show that while H-like line fluxes can be accurately predicted, fluxes of lower-Z He-like lines are significantly underestimated. This discrepancy suggests that some extra heating mechanism plays an important role at the base of the accretion column, where cooler ions form. We thus explored more complex cooling models including the change of gravitational potential with height in the accretion column and a magnetic dipole geometry. None of these modifications to the standard cooling flow model are able to reproduce the observed line ratios. While a cooling flow model with subsolar (0.1 $odot$) abundances is able to reproduce the line ratios by reducing the cooling rate at temperatures lower than $sim 10^{7.3}$ K, the predicted line-to-continuum ratios are much lower than observed. We discuss and discard mechanisms such as photoionization, departures from constant pressure, resonant scattering, different electron-ion temperatures, and Compton cooling. [Abridged]
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