Principal argument of $z=\frac{i-1}{i\left(1\cos \frac{2\pi }{7}\right)+\sin \frac{2\pi }{7}}$ is 

$i\left(1-\cos \frac{2\pi }{7}\right)+\sin \frac{2\pi }{7}$

$\begin{array}{r}=2i{\sin }^2\frac{\pi }{7}+2\sin \frac{\pi }{7}\cos \frac{\pi }{7}\\ =2\sin \left(\frac{\pi }{7}\right)\left[\cos \frac{\pi }{7}+i\sin \frac{\pi }{7}\right]\end{array}$

Also , $i-1=\sqrt{2}\left(-\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\right)$

$=\sqrt{2}\left(\cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4}\right)$

$\therefore z=\frac{\sqrt{2}}{2\sin \left(\frac{\pi }{7}\right)}\left[\cos \left(\frac{3\pi }{4}-\frac{\pi }{7}\right)+i\sin \left(\frac{3\pi }{4}-\frac{\pi }{7}\right)\right]$

$=\frac{\sqrt{2}}{2\sin \left(\frac{\pi }{7}\right)}\left[\cos \left(\frac{17\pi }{28}\right)+i\sin \left(\frac{17\pi }{28}\right)\right]$

Thus , $\arg \left(z\right)=\frac{17\pi }{28}$.