Type Iax supernovae (SNe Iax) are the largest known class of peculiar white dwarf SNe, distinct from normal Type Ia supernovae (SNe Ia). The unique properties of SNe Iax, especially their strong photospheric lines out to extremely late times, allow us to model their optical spectra and derive the physical parameters of the long-lasting photosphere. We present an extensive spectral timeseries, including 21 new spectra, of SN Iax 2014dt from +11 to +562 days after maximum light. We are able to reproduce the entire timeseries with a self-consistent, nearly unaltered deflagration explosion model from Fink et al. using TARDIS, an open source radiative-transfer code. We find that the photospheric velocity of SN 2014dt slows its evolution between +64 and +148 days, which closely overlaps the phase when we see SN 2014dt diverge from the normal spectral evolution of SNe Ia (+90 to +150 days). The photospheric velocity at these epochs, ~400-1000km/s, may demarcate a boundary within the ejecta below which the physics of SNe Iax and normal SNe Ia differ. Our results suggest that SN 2014dt is consistent with a weak deflagration explosion model that leaves behind a bound remnant and drives an optically thick, quasi-steady-state wind creating the photospheric lines at late times. The data also suggest that this wind may weaken at epochs past +450d, perhaps indicating a radioactive power source that has decayed away.