In case $d$ increases linearly in $k$, say $k=\alpha d$, you can compute the limiting spectral measure of your matrix (you are dealing with the product of two wishart matrices, so you can compute the limit e.g. by computing moments, or better using free probability). Call the limit distribution $\rho$. Then the statistics you are after is $\sqrt{\int_{F^{-1}(\alpha)}^\infty x^2 dF(x)}$. If you want estimates on the error, you can use concentration inequalities for the spectral measure (since the top singular value concentrates as well).

In case $d$ increases linearly in $k$, say $k=\alpha d$, you can compute the limiting spectral measure of your matrix (you are dealing with the product of two wishart matrices, so you can compute the limit e.g. by computing moments, or better using free probability). Call the limit distribution $F$. Then the statistics you are after is $\sqrt{\int_{F^{-1}(\alpha)}^\infty x^2 dF(x)}$. If you want estimates on the error, you can use concentration inequalities for the spectral measure (since the top singular value concentrates as well).