  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]
