Microlensing has proved an effective probe of the structure of the innermost regions of quasars and an important test of accretion disk models. We present light curves of the lensed quasar HE 0435-1223 in the R band and in the ultraviolet (UV), and consider them together with X-ray light curves in two energy bands that are presented in a companion paper. Using a Bayesian Monte Carlo method, we constrain the size of the accretion disk in the rest-frame near- and far-UV, and constrain for the first time the size of the X-ray emission regions in two X-ray energy bands. The R-band scale size of the accretion disk is about 10^15.23^ cm (~23r_g_), slightly smaller than previous estimates, but larger than would be predicted from the quasar flux. In the UV, the source size is weakly constrained, with a strong prior dependence. The UV to R-band size ratio is consistent with the thin disk model prediction, with large error bars. In soft and hard X-rays, the source size is smaller than ~10^14.8^ cm (~10r_g_) at 95% confidence. We do not find evidence of structure in the X-ray emission region, as the most likely value for the ratio of the hard X-ray size to the soft X-ray size is unity. Finally, we find that the most likely value for the mean mass of stars in the lens galaxy is ~0.3 M_{sun}_, consistent with other studies.