We present deep Chandra observations and Spitzer Space Telescope infrared ( IR ) spectroscopy of the shell in the composite supernova remnant ( SNR ) Kes 75 ( G29.7-0.3 ) . The remnant is composed of a central pulsar wind nebula and a bright partial shell in the south that is visible at radio , IR , and X-ray wavelengths . The X-ray emission can be modeled by either a single thermal component with a temperature of \sim 1.5 keV , or with two thermal components with temperatures of 1.5 and 0.2 keV . Previous studies suggest that the hot component may originate from reverse-shocked SN ejecta . However , our new analysis shows no definitive evidence for enhanced abundances of Si , S , Ar , Mg , and Fe , as expected from supernova ( SN ) ejecta , or for the IR spectral signatures characteristic of confirmed SN condensed dust , thus favoring a circumstellar or interstellar origin for the X-ray and IR emission . The X-ray and IR emission in the shell are spatially correlated , suggesting that the dust particles are collisionally heated by the X-ray emitting gas . The IR spectrum of the shell is dominated by continuum emission from dust with little , or no line emission . Modeling the IR spectrum shows that the dust is heated to a temperature of \sim 140 K by a relatively dense , hot plasma , that also gives rise to the hot X-ray emission component . The density inferred from the IR emission is significantly higher than the density inferred from the X-ray models , suggesting a low filling factor for this X-ray emitting gas . The total mass of the warm dust component is at least 1.3 \times 10 ^ { -2 } \ > M _ { \odot } , assuming no significant dust destruction has occurred in the shell . The IR data also reveal the presence of an additional plasma component with a cooler temperature , consistent with the 0.2 keV gas component . Our IR analysis therefore provides an independent verification of the cooler component of the X-ray emission . The complementary analyses of the X-ray and IR emission provide quantitative estimates of density and filling factors of the clumpy medium swept up by the SNR .