Context. Most analyses of the infrared emission of Saturn's rings and icy satellites have considered pure water ice as the constituent of regolith and particle surfaces. Visual and near-infrared observations have shown, however, that darkening and reddening contaminants are present at a fraction level of a few percent. In the spectral domain 10-2000 cm(-1), water ice becomes transparent in a few windows, which in particular causes the roll-off of emissivity of icy surfaces that is observed below 50 cm(-1). Their emissivity there may be affected by these contaminants. Aims. We present a quantitative global sensitivity analysis of a hybrid Mie-Hapke model to evaluate the influence of regolith properties and contaminant fraction on the infrared emissivity of icy rings or moons over this spectral range. Methods. A hybrid Mie-Hapke model of the hemispherical emissivity epsilon(h)*(W-n) was made, including various diffraction correction and mixing types with tholins or amorphous carbon grains, or grain size distributions and some anisotropy in emission. A Sobol global sensitivity analysis provided quantitative levels of importance for these factors versus wave number w(n). Results. Given the a priori uncertainties, the most important factor acting on epsilon*h(W-n) remains the size distribution of regolith grains and the average anisotropy factor xi. For W-n> 50 cm(-1), xi, the power-law index p and the minimum a(min) of the size distribution are most influential. In windows of water-ice transparency (10-50, 300-600, and 900-1300 cm(-1)), the emissivity is also sensitive, but to a lesser extent, to the maximum grain size a(max) and the fraction f of contaminants, if mixed at the molecular level. Conclusions. This model provides a self-consistent tool for interpreting multi-modal observations of the thermal emission from icy surfaces. It also offers interesting insights into recent mid-infrared observations of Saturn's rings and Jupiter's moon Ganymede by the JWST-MIRI instrument.