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+0.02Z⊙ . 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(ξ ion) = 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.