First determine the damping ratio $\zeta$ and natural frequency $\omega$ of the closed loop poles.
The general characteristic equation is $s^2+2 \zeta s \omega +\omega ^2$. For the desired pole locations the characteristic equation is $(s+10 -8.83 i) (s+10 +8.83 i)$. Equate the coefficients and solve for $\zeta$ and $\omega$.
Now draw lines from the origin to the desired closed-loop poles at $-\zeta \omega \pm i \sqrt{1-\zeta ^2} \omega $. The lines must intersect with the root-locus plot to get a feasible $K$. The $K$ value at which the intersection occurs is the value you are looking for.
You did not provide the transfer function, so I made one up. Your analysis will be something like this.

Update:
The solution according to your professor's logic:
The characteristic equation is $$1 + K G(s) = 0$$
This implies $$K = \left| \frac{1}{G(s)}\right|$$
From the figure we know that $G(s)$ has two finite poles and one finite zero. So write $$G(s) = \frac{s-z}{(s-p_1)(s-p_2)}$$
The closed loop pole needs to be at $s = -10 +8.83 i$ (the conjugate will give the same result). Thus:
$$K = \left|\frac{\left(-10+8.83 i-p_1\right) \left(-10+8.83 i-p_2\right)}{-10+8.83i-z}\right|$$
But you again need to read off the values of the open-loop zero and open-loop poles from the figure to get $z$, $p_1$, and $p_2$.