Finding desired closed-loop poles from root locus
The issue you have is you have plotted the natural frequency \$\omega_n\$ of the system not a design requirement for settling time. You want to plot a vertical line at $$ s \simeq - \dfrac{3.9}{t_s} $$

If you have Matlab you can also accomplish this with the SISO tool. Add a design requirement for settling time and you will get a vertical line. If the complex pole-pair exists towards neg-infinity in the left-hand plane, you have satisfied the settling time requirement. 

Sample root locus plot below. 


Here any complex-pole pair that exists in the white space satisfies design requirements for overshoot and settling time.