Gaussian width of sparse balls The Gaussian width of a set $T\subset \mathbb{R}^n$ is defined as,
$$
G(T) = E\left[\sup_{\theta \in T} \sum_{i=1}^n \theta_i W_i\right],
$$
where, $\mathbf{W}=(W_1,\ldots,W_n)$ is a sequence of i.i.d. $N(0,1)$ random variables. I am interested in finding the value of $G(T)$ for 
$$
T(s) \equiv \{\theta\in\mathbb{R}^n: {\|\theta\|}_0 \leq s,{\|\theta\|}_2\leq 1\},
$$
the set of all $s$-sparse vectors within the unit ball, with $s\in\{1,\ldots,n\}$. This is an exercise problem in Wainwright's book on HD-Statistics. I have been able to show,
$$
G(T(s)) = E\max_{|S|=s} {\|\mathbf{W}_S\|}_2,
$$
and $S$ is a subset of $\{1,\ldots,n\}$, with cardinality $|S| = s$. Here the subscript $S$ denotes the components of $\mathbf{W}$ corresponding to $S$.
Then, using Gaussian concentration inequality and the union bound, I can get,
$$
P\left(\max_{|S|=s}{\|\mathbf{W}_S\|}_2 \geq \sqrt{s} + t\right)\leq \binom{n}{s} \exp\{-t^2/2\},\ \text{for all $t>0$.}
$$
I can use the bound,
$$
\binom{n}{s}\leq {\left(ne/s\right)}^s, \ \text{for all $s=1,\ldots,n$.}
$$
Finally, I need to integrate to obtain the bound on the expectation. I am unable to do it to get the desired upper bound (of the order),
$$
K\sqrt{s\log(en/s)},\ \text{where $K$ is some constant.}
$$
Any ideas would be helpful!

Notation: $C$ below denotes (possibly different) absolute constants.
Recall that for $N$ sub-gaussian variable $X_i$ (independence not required) with $\max_i \| X_i\|_{\psi_2}\le K$, $E \max_{i\le N} X_i \le CK \sqrt{\log{N}}.$
For our problem, max in $E \max_{|S| \le s} |W_S|$ enumerates over $N:=\sum_{k=1}^s \binom{n}{k}$ different subsets of ${1,\dots,n}$. Also use Gaussian concentration inequality we obtain $\max_{|S|\le s} \| W_S-\sqrt{|S|}\|_{\psi_2}\le C$.
So we have $$E \max_{|S|\le s} (|W_S|-\sqrt{|S|}) \le C \sqrt{\log(\sum_{k=1}^s \binom{n}{k})} \le C \sqrt{s\log(en/s)}$$
where we used $\sum_{k=1}^s \binom{n}{k} \le (\frac{ne}{s})^s.$
Finally, this implies that (using $\sqrt{|S|}\le \sqrt{s}$ and move $\sqrt{s}$ to RHS)
$$E \max_{|S| \le s} W_S \le \sqrt{s}+C\sqrt{s\log(en/s)}\le C \sqrt{s\log(en/s)}$$
by $n>s$.
