The increase in the number of Type Ia supernovae (SNe Ia) has demonstrated that the population shows greater diversity than has been assumed in the past. The reasons (e.g. parent population, explosion mechanism) for this diversity remain largely unknown. We investigated a sample of SNe Ia near-infrared light curves and correlated the phase of the second maximum with the bolometric peak luminosity. The peak bolometric luminosity is related to the time of the second maximum (relative to the B light curve maximum) as follows: L-max(10(43) erg s(-1)) = (0.039 +/- 0.004) x t(2)(J)(days) + (0.013 +/- 0.106). Ni-56 masses can be derived from the peak luminosity based on Arnett's rule, which states that the luminosity at maximum is equal to the instantaneous energy generated by the nickel decay. We checked this assumption against recent radiative-transfer calculations of Chandrasekhar-mass delayed detonation models and find this assumption is valid to within 10% in recent radiative-transfer calculations of Chandrasekhar-mass delayed detonation models. The L-max vs. t(2) relation is applied to a sample of 40 additional SNe Ia with significant reddening (E(B - V) \textgreater 0.1 mag), and a reddening-free bolometric luminosity function of SNe Ia is established. The method is tested with the Ni-56 mass measurement from the direct observation of gamma-rays in the heavily absorbed SN 2014J and found to be fully consistent. Super-Chandrasekhar-mass explosions, in particular SN 2007if, do not follow the relations between peak luminosity and second IR maximum. This may point to an additional energy source contributing at maximum light. The luminosity function of SNe Ia is constructed and is shown to be asymmetric with a tail of low-luminosity objects and a rather sharp high-luminosity cutoff, although it might be influenced by selection effects.