Galaxy mergers are key events in galaxy evolution, often causing massive starbursts and fueling active galactic nuclei (AGNs). In these highly dynamic systems, it is not yet precisely known how much starbursts and AGNs, respectively, contribute to the total luminosity, at what interaction stages they occur, and how long they persist. Here we estimate the fraction of the bolometric infrared (IR) luminosity that can be attributed to AGNs by measuring and modeling the full ultraviolet to far-IR spectral energy distributions (SEDs) in up to 33 broad bands for 24 merging galaxies with the Code for Investigating Galaxy Emission. In addition to a sample of 12 confirmed AGNs in late-stage mergers, found in the Infrared Astronomical Satellite Revised Faint Source Catalog, our sample includes a comparison sample of 12 galaxy mergers from the Spitzer Interacting Galaxies Survey, mostly early stage. We also perform SED modeling of merger simulations to validate our methods, and we supplement the SEDs with mid-IR spectra of diagnostic lines obtained with Spitzer's InfraRed Spectrograph. The estimated AGN contributions to the IR luminosities vary from system to system from 0 per cent up to ~91 per cent but are significantly greater in the later-stage, more luminous mergers, consistent with what is known about galaxy evolution and AGN triggering.