We present the Apache Point Observatory BG40 broadband and simultaneous Gemini $r$-band and $i$-band high-speed follow-up photometry observations and analysis of the 40.5 minute period eclipsing detached double-degenerate binary SDSS J082239.54$+$304857.19. Our APO data spans over 318 days and includes 13 primary eclipses, from which we precisely measure the systems orbital period and improve the time of mid-eclipse measurement. We fit the light curves for each filter individually and show that this system contains a low-mass DA white dwarf with radius $R_A=0.031pm0.006~{rm R_odot}$ and a $R_B=0.013pm0.005~{rm R_odot}$ companion at an inclination of $i=87.7pm0.2^circ$. We use the best-fitting eclipsing light curve model to estimate the temperature of the secondary star as $T_{rm eff}=5200pm100~{rm K}$. Finally, while we do not record significant offsets to the expected time of mid-eclipse caused by the emission of gravitational waves with our 1-year baseline, we show that a $3sigma$ significant measurement of the orbital decay due to gravitational waves will be possible in 2023, at which point the eclipse will occur about $8$ seconds earlier than expected.
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