The late-time spectra of Type Ia supernovae (SNe Ia) are powerful probes of the underlying physics of their explosions. We investigate the late-time optical and near-infrared spectra of seven SNe Ia obtained at the VLT with XShooter at > 200 d after explosion. At these epochs, the inner Fe-rich ejecta can be studied. We use a line-fitting analysis to determine the relative line fluxes, velocity shifts, and line widths of prominent features contributing to the spectra ([Fe II], [Ni II], and [Co III]). By focusing on [Fe II] and [Ni II] emission lines in the ~7000-7500 Å region of the spectrum, we find that the ratio of stable [Ni II] to mainly radioactively produced [Fe II] for most SNe Ia in the sample is consistent with Chandrasekharmass delayed-detonation explosion models, as well as sub-Chandrasekhar mass explosions that have metallicity values above solar. The mean measured Ni/Fe abundance of our sample is consistent with the solar value. The more highly ionized [Co III] emission lines are found to be more centrally located in the ejecta and have broader lines than the [Fe II] and [Ni II] features. Our analysis also strengthens previous results that SNe Ia with higher Si II velocities at maximum light preferentially display blueshifted [Fe II] 7155 Å lines at late times. Our combined results lead us to speculate that the majority of normal SN Ia explosions produce ejecta distributions that deviate significantly from spherical symmetry.