I do not know whether the following computation may be carried on to give a simple formula. May be is there a reference giving the value of your integral, but I do not know it.
By using the elementary divisor theorem, you are reduced to sum a series whose terms are of the form
$$
\int_{K.t_\lambda .K} \vert {\rm det} X\vert^s\, dX
$$
where $K={\rm GL}(n,{\mathcal O})$, $\lambda$ runs over the partition of $n$ of the form $k_1 \geqslant k_2 \geqslant \cdots \geqslant k_n >0$ and $t_\lambda ={\rm diag}(\pi^{k_1} ,...,\pi^{k_n})$.
You may rewrite this integral as
$$
\int_{K.t_\lambda .K} \vert {\rm det} X\vert^{s+1}\, \frac{dX}{\vert {\rm det} X\vert}
$$
and notice that $dX/\vert {\rm det} X\vert$ is a Haar measure on ${\rm GL}(n,F)$.
The integral reduces to
$$
q^{-(s+1)(k_1 +\cdots + k_n)}\, \mu (K t_\lambda K)
$$
and you are thus reduced to compute the volume of $K t_\lambda K$ with respect to the Haar measure on ${\rm GL}(n,F)$.
This volume is given by
$$
W (1/q)/W_\lambda (1/q) \, \cdot \, \delta (t_{\lambda })\, \cdot \, \mu (K)
$$
where :
-- $W$ is the Poincaré polynomial of the spherical Weyl ${\mathfrak S}_n$ group of ${\rm GL}(n)$,
-- $W_\lambda$ is the Poincaré polynomial of the centralizer of $\lambda$ in ${\mathfrak S}_n$,
-- $\delta$ is the modulus function of the standard Borel subgroup of ${\rm GL}(n,F)$.
A reference for this volume formula is : Proposition (3.2.15), p.44, of Spherical Functions on a Group of p-adic Type, by Macdonald, Publication of the Ramanujan Institut Number 2.
A reference for the volume of this double class
