The Didymos binary asteroid was the target of the Double Asteroid Redirection Test (DART) mission, which intentionally impacted Dimorphos, the smaller member of the binary system. We used the Near-Infrared Spectrograph and Mid-Infrared Instrument instruments on JWST to measure the 0.6-5 and 5-20 μm spectra of Didymos approximately two months after the DART impact. These observations confirm that Didymos belongs to the S asteroid class and is most consistent with LL chondrite composition, as was previously determined from its 0.6-2.5 μm reflectance spectrum. Measurements at wavelengths >2.5 μm show Didymos to have thermal properties typical for an S-complex asteroid of its size and to be lacking absorptions deeper than ~2% due to OH or H2O. Didymos' mid-infrared emissivity spectrum is within the range of what has been measured on S-complex asteroids observed with the Spitzer Space Telescope and is most consistent with emission from small (<25 μm) surface particles. We conclude that the observed reflectance and physical properties make the Didymos system a good proxy for the type of ordinary chondrite asteroids that cross near-Earth space, and a good representative of likely future impactors.