Using the SEDs of the weak AGNs 35 LINERs presented in a companion paper, we assess whether photoionization by the weak AGN can power the emission-line luminosities measured through the large (few-arcsecond) apertures used in ground-based spectroscopic surveys. Spectra taken through such apertures are used to define LINERs as a class and constrain non-stellar photoionization models for LINERs. Therefore, our energy budget test is a self-consistency check of the idea that the observed emission lines are powered by an AGN. We determine the ionizing luminosities and photon rates by integrating the observed SEDs and by scaling a template SED. Even if all ionizing photons are absorbed by the line-emitting gas, more than half of our LINERs suffer from a deficit of ionizing photons. In 1/3 of LINERs the deficit is severe. If only 10% of the ionizing photons are absorbed by the gas, there is an ionizing photon deficit in 85% of LINERs. We disfavor the possibility that additional electromagnetic power, either obscured or emitted in the unobservable far-UV band, is available from the AGN. We consider other power sources such as mechanical heating by compact jets and photoionization by young or old stars. Photoionization by young stars may be important in a small fraction of cases. Mechanical heating provides enough power in most cases but it is not clear how this power is transferred to the emission-line gas. Photoionization by post-AGB stars is an important power source; it provides more ionizing photons that the AGN in more than half of the LINERs and enough ionizing photons to power the emission lines in 1/3 of the LINERs. It appears likely that the emission-line spectra of LINERs obtained from the ground include the sum of emission from different regions where different power sources dominate.
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