JWST spectroscopy has revolutionized our understanding of galaxies in the early Universe. Covering wavelengths up to 5.3 μm, NIRSpec can detect rest-frame optical Hα emission lines out to z = 7 and [O iii] to z = 9.5. Observing these lines in more distant galaxies requires longer wavelength spectroscopy with MIRI. Here we present MIRI Medium Resolution Spectrograph integral field unit observations of the lensed galaxy merger MACS0647-JD at z = 10.165. With exposure times of 4.2 hr in each of two bands (SHORT and LONG), we detect Hα at 9σ, [O iii] λ5008 at 11σ, and [O iii] λ4960 at 3σ. Combined with previously reported NIRSpec spectroscopy that yielded seven emission lines including the auroral line [O iii] λ4363, we present the first direct metallicity measurement of a z > 10 galaxy: 12 + log ( O / H ) = 7.79 ± 0.09 , or 0.13 − 0.03 + 0.02 Z ⊙ . This is similar to galaxies at z ∼ 4-9 with direct metallicity measurements, though higher than expected given the high specific star formation rate log(sSFR/yr−1) = −7.4 ± 0.3. We further constrain the ionization parameter log ( U ) = −1.9 ± 0.1, ionizing photon production efficiency log(ξion) = 25.3 ± 0.1, and SFR = 5.0 ± 0.6 M⊙ yr−1 within the past 10 Myr. These observations demonstrate the combined power of JWST NIRSpec and MIRI for studying galaxies in the first 500 million years.