The two-loop renormalization constants $Z_m$ and $Z_1=Z_2$ of Feynman gauge quantum electrodynamics (QED) in the on-shell scheme are computed using dimensional regularization. Analytic results are obtained for these constants. In addition, analytic expressions are obtained for their expansion coefficients in the dimensional regulation parameter $ϵ$ from order $1/{ϵ}^2$ through order ${ϵ}^5$. The high order expansion coefficients will be needed for future multi-loop calculations in QED and will provide a useful check for multi-loop calculations in QCD.