It can be proved by induction on $n$.
Base case $n=1$.
The step follows since gradient exponent is locally Lipschitz and it maps $T_p(\partial X)=\mathrm{Cone}[\Sigma_p(\partial X)]$ to a neighborhood of $p$ in $\partial X$.
It proves that $\partial X$ has finite $(n-1)$-Hausdorff measure.

The differential of gradient exponent at the origin is the identity map.
It follows that a small cube $\square$ around a regular point in $T_p(\partial X)$ is mapped amost isometrically; in particular the opposite faces of $\square$ go to sets on positive distance. By Besicovitch inequality, the image of $\square$ has positive $(n-1)$-Hausdorff measure.

The same proof shows that an $m$-dimensional extremal subset in a compact Alexandrov space has finite positive $m$-dimesnional Hausdoeff measure.