There are $11$ terms in the expansion $\Rightarrow 6^{th}$ term is the middle term

${\mathrm{T}}_6={}_{}{}^{10}C_5{\left(\frac{1}{x}\right)}^5{\left(x\sin x\right)}^5$
$\Rightarrow {}_{}{}^{10}C_5{\left(\sin x\right)}^5=\frac{63}{8}$
$\Rightarrow \frac{10\times 9\times 8\times 7\times 6}{5\mathrm{}\times 4\times 3\times 2}{\left(\sin x\right)}^5=\frac{63}{8}\mathrm{}$
$\Rightarrow {\left(\sin x\right)}^5=\frac{63}{8}\times \frac{5\mathrm{}\times 4\times 3\times 2}{10\times 9\times 8\times 7\times 6}$
$\Rightarrow \sin x=\frac{1}{2}$

Hence, there are $2$ values of $x$ in $\left[0,2\mathrm{\pi }\right]$