The distribution of the absorption line broadening observed in the Ly alpha forest carries information about the temperature, T, and widths, lambda(F), of the filaments in the intergalactic medium (IGM), and the background hydrogen photoionization rate, Gamma(HI). In this work, we present and test a new method for inferring T and lambda(F) and Gamma HI from combining the distribution of the absorption line broadening and the median flux. The method accounts for any underlying degeneracies. We apply our method to mock spectra from the reference model of the EAGLE cosmological simulation, and we demonstrate that we are able to reconstruct the IGM properties.