The gravitationally lensed quasar HE 1104-1805 has been observed at a variety of wavelengths ranging from the mid-infrared to X-ray for nearly 20yr. We combine flux ratios from the literature, including recent Chandra data, with new observations from the SMARTS telescope and Hubble Space Telescope, and use them to investigate the spatial structure of the central regions using a Bayesian Monte Carlo analysis of the microlensing variability. The wide wavelength coverage allows us to constrain not only the accretion disk half-light radius r_1/2_, but the power-law slope {xi} of the size-wavelength relation r_1/2_{prop.to}{lambda}^{xi}^. With a logarithmic prior on the source size, the (observed-frame) R-band half-light radius log (r_1/2_/cm) is 16.0_-0.4_^+0.3^, and the slope {xi} is 1.0_-0.56_^+0.30^. We put upper limits on the source size in soft (0.4-1.2 keV) and hard (1.2-8keV) X-ray bands, finding 95% upper limits on log (r_1/2_/cm) of 15.33 in both bands. A linear prior yields somewhat larger sizes, particularly in the X-ray bands. For comparison, the gravitational radius, using a black hole mass estimated using the H{beta} line, is log(r_g_/cm)=13.94. We find that the accretion disk is probably close to face-on, with cos(i)=1.0 being four times more likely than cos(i)=0.5. We also find probability distributions for the mean mass of the stars in the foreground lensing galaxy, the direction of the transverse peculiar velocity of the lens, and the position angle of the projected accretion disk's major axis (if not face-on).