Some supernovae (SNe) may be powered by the interaction of the SN ejecta with a large amount of circumstellar matter (CSM). However, quantitative estimates of the CSM mass around such SNe are missing when the CSM material is optically thick. Specifically, current estimators are sensitive to uncertainties regarding the CSM density profile and the ejecta velocity. Here we outline a method to measure the mass of the optically thick CSM around such SNe. We present new visible-light and X-ray observations of SN 2010jl (PTF 10aaxf), including the first detection of an SN in the hard X-ray band using NuSTAR. The total radiated luminosity of SN 2010jl is extreme-at least 9 × 1050 erg. By modeling the visible-light data, we robustly show that the mass of the circumstellar material within ~1016 cm of the progenitor of SN 2010jl was in excess of 10 M⊙. This mass was likely ejected tens of years prior to the SN explosion. Our modeling suggests that the shock velocity during shock breakout was ~6000 km s-1, decelerating to ~2600 km s-1 about 2 yr after maximum light. Furthermore, our late-time NuSTAR and XMM spectra of the SN presumably provide the first direct measurement of SN shock velocity 2 yr after the SN maximum light-measured to be in the range of 2000-4500 km s-1 if the ions and electrons are in equilibrium, and ≳ 2000 km s-1 if they are not in equilibrium. This measurement is in agreement with the shock velocity predicted by our modeling of the visible-light data. Our observations also show that the average radial density distribution of the CSM roughly follows an r-2 law. A possible explanation for the ≳ 10 M⊙ of CSM and the wind-like profile is that they are the result of multiple pulsational pair instability events prior to the SN explosion, separated from each other by years.