Consider some scalars $d_1\leq \dots\leq d_k$ and let $D = \text{diag}(d_i)\in\mathbb{R}^{k\times k}$. Assume that $X\in \mathbb{R}^{k \times k}$ is a random matrix of i.i.d. normal entries (i.e. $X_{ij} \sim \mathcal{N}(0,1)$). I want to estimate
$$ f(D) = \mathbb{E}\left[\frac{\|DX\|^2}{\|X\|^2}\right] $$ where $\|\cdot\|$ is the spectral norm (i.e. the largest singular value).
I have not been able to come up with better inequalities than the trivial $d_1^2\leq f(D)\leq d_k^2$. I am not really sure that there is a closed form expression for $f$. With numerical simulations, even when $k=2$, I only have a rough approximation $f(\text{diag}(1, t))\simeq \frac12 t^2 + 0.57$.
Any help is greatly appreciated.