  A common approach to studying $\beta$-delayed proton emission is to measure
the energy of the emitted proton and corresponding nuclear recoil in a
double-sided silicon-strip detector (DSSD) after implanting the $\beta$-delayed
proton emitting ($\beta$p) nucleus. However, in order to extract the
proton-decay energy, the measured energy must be corrected for the additional
energy implanted in the DSSD by the $\beta$-particle emitted from the $\beta$p
nucleus, an effect referred to here as $\beta$-summing. We present an approach
to determine an accurate correction for $\beta$-summing. Our method relies on
the determination of the mean implantation depth of the $\beta$p nucleus within
the DSSD by analyzing the shape of the total (proton + recoil + $\beta$) decay
energy distribution shape. We validate this approach with other mean
implantation depth measurement techniques that take advantage of energy
deposition within DSSDs upstream and downstream of the implantation DSSD.
