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We constrain the role of different SN Ia channels in the chemical enrichment of the Galaxy by studying the abundances of nickel in Galactic stars. We investigate four different SN Ia sub-classes, including the classical single-degenerate near-Chandrasekhar mass SN Ia, the fainter SN Iax systems associated with He accretion from the companion, as well as two sub-Ch mass SN Ia channels. The latter include the double-detonation of a white dwarf accreting helium-rich matter and violent white dwarf mergers. NLTE models of Fe and Ni are used in the abundance analysis. In the GCE models, we include new delay time distributions arising from the different SN Ia channels, as well as recent yields for core-collapse supernovae and AGB stars. The data-model comparison is performed using a Markov chain Monte Carlo framework that allows us to explore the entire parameter space allowed by the diversity of explosion mechanisms and the Galactic SN Ia rate, taking into account the uncertainties of the observed data. We show that NLTE effects have a non-negligible impact on the observed [Ni/Fe] ratios in the Galactic stars. The NLTE corrections to Ni abundances are not large, but strictly positive, lifting the [Ni/Fe] ratios by ~+0.15 dex at [Fe/H] =-2. We find that that the distributions of [Ni/Fe] in LTE and in NLTE are very tight, with a scatter of < 0.1 dex at all metallicities, supporting earlier work. In LTE, most stars have scaled-solar Ni abundances, [Ni/Fe] = 0, with a slight tendency for sub-solar [Ni/Fe] ratios at lower [Fe/H]. In NLTE, however, we find a mild anti-correlation between [Ni/Fe] and metallicity, and a slightly elevated [Ni/Fe] ratios at [Fe/H] < -1.0. The NLTE data can be explained by the GCE models calculated with a substantial, ~ 75%, fraction of sub-Ch SN Ia.