The first two moments of $X_i$ are given, $\mathbb{E}[X_i]=\mu_i$, $\mathbb{E}[X_i^2]=\sigma_i^2+\mu_i^2$. We also need the fourth moment, $\mathbb{E}[X_i^4]=\tau_i^4$.
Then with $Y \equiv \sqrt{\sum_{i=1}^k(\frac{ X_i}{\sigma_i})^2}$, we have $$\mathbb{E}[Y^2]=M=\sum_{i=1}^k\bigl[1+(\mu_i/\sigma_i)^2\bigr],$$ $$\text{var}\,[Y^2]=V=\sum_{i=1}^k \bigl[(\tau_i/\sigma_i)^4-\bigl(1+(\mu_i/\sigma_i)^2\bigr)^2\bigr].$$ Both $M$ and $V$ are assumed to scale linearly with $k$. For large $k$ the distribution of $Y^2$ tends to a Gaussian with mean $M$ and variance $V$. We now follow the same steps as in https://mathoverflow.net/a/453690/11260 : to leading order in $1/k$ one has $$\mathbb{E}[Y] =M^{1/2}+{\cal O}(k^{-1/2}),$$ $$\text{var}\,[Y]=\frac{V}{4 M}+{\cal O}(k^{-1}).$$ So the mean of $Y$ scales as $\sqrt{k}$ while the variance tends to a constant in the limit $k\rightarrow\infty$.