Let $(e_1,\dots,e_k)$ denote the standard basis in $\mathbb{R}^k$.
Then
$$\frac1{\sqrt n}\,(X_1-np_1,\dots,X_k-np_k)\overset{D}=\frac1{\sqrt n}\,\sum_{j=1}^nV_j=:V,
$$
where $\overset{D}=$ stands for the equality in distribution,
$V_j:=U_j-\mathbb{E}U_j$, and the $U_j$'s are iid random vectors in $\mathbb{R}^k$ such that $\mathbb{P}(U_j=e_\ell)=p_\ell$ for $\ell=1,\dots,k$.
Let $W\sim N(0,\Sigma)$, where $\Sigma$ is the covariance matrix of $V_1$.

We shall show more than requested:
$$(0)\qquad\big|\,\mathbb{E}\|V\|-\mathbb{E}\|W\|\,\big|<0.99 n^{-1/6};
$$
the latter bound is uniform, not only in the $p_\ell$'s, but also in $k$.

By Corollary 2.1 and formula (1.10) in [1],
$$(1)\qquad\big|\,\mathbb{E}\|V\|-\mathbb{E}\|W\|\,\big|
\le\sqrt{\frac2\pi}\,\int_{\mathbb{R}^k}\gamma_k(du)
\int_0^\infty\frac{dt}{t^2}|\mathbb{E}e^{itu\cdot V}-\mathbb{E}e^{itu\cdot W}|,
$$
where $\gamma_k$ is the standard Gaussian measure on $\mathbb{R}^k$.

Using inequality (13) in [2] and writing $\int_0^{|b|}\frac{a^2}2\,\exp\big(\frac{a^2}2\big)\,da
\le e^{b^2/2}\frac{|b|^3}6$ for real $b$, for all $u\in\mathbb{R}^k$ and $t\in\mathbb{R}$ we have
$$(2)\qquad|\mathbb{E}e^{itu\cdot V}-\mathbb{E}e^{itu\cdot W}|
\le c|t|^3\delta_n(u),\quad\text{where}\quad\delta_n(u):=\frac{\mathbb{E}|u\cdot V_1|^3}{\sqrt n}
$$
and $c:=1/(6\sqrt{2\pi})$.
Let $A:=\big(2/(c\delta_n(u))\big)^{1/3}$.
Using (2) for $t\in(0,A)$ and the trivial inequality $|\mathbb{E}e^{itu\cdot V}-\mathbb{E}e^{itu\cdot W}|\le2$ for $t\ge A$, for the integral $\int_0^\infty$ in (1) one has
$$\int_0^\infty\le\int_0^A t\,dt\,c\delta_n(u)
+2\int_A^\infty\frac{dt}{t^2}
=3\times2^{-1/3}(c\delta_n(u))^{1/3}.
$$
So, by (1), with $c_1:=\sqrt{\frac2\pi}\,\times3\times2^{-1/3}c^{1/3}$,
$$(3)\qquad\big|\,\mathbb{E}\|V\|-\mathbb{E}\|W\|\,\big|
\le c_1 \int_{\mathbb{R}^k}\gamma_k(du)\delta_n(u)^{1/3}
\le c_1 \Big(\int_{\mathbb{R}^k}\gamma_k(du)\delta_n(u)\Big)^{1/3}.
$$
Next, by part (vi) of Theorem 2.3 of [3], $\mathbb{E}|u\cdot V_1|^3\le1.32\mathbb{E}|u\cdot U_1|^3
=1.32\sum_{\ell=1}^k p_\ell|u_\ell|^3$, where $u=(u_1,\dots,u_k)\in\mathbb{R}^k$.
So,
$$\sqrt n\,\int_{\mathbb{R}^k}\gamma_k(du)\delta_n(u)
=\int_{\mathbb{R}^k}\gamma_k(du)\mathbb{E}|u\cdot V_1|^3
\le1.32\int_{\mathbb{R}^k}\gamma_k(du)\mathbb{E}|u\cdot U_1|^3
$$
$$=1.32\sum_{\ell=1}^k p_\ell\int_{\mathbb{R}^k}\gamma_k(du)|u_\ell|^3
=1.32\sum_{\ell=1}^k p_\ell\int_{\mathbb{R}}\gamma_1(du_1)|u_1|^3
=1.32\times2\sqrt{\frac2\pi}.
$$
Now (0) follows from (3).

Similarly one can obtain bounds on the rate of convergence for $\mathbb{E}\|V\|^p$ for $p$ other than $1$.

[1] http://arxiv.org/abs/1506.00537 ; a version of this is to appear In The American Mathematical Monthly.

[2] http://arxiv.org/abs/0912.0726

[3] http://arxiv.org/abs/1111.2622 ; published at http://link.springer.com/chapter/10.1007%2F978-3-0348-0490-5_6