Trouble with meeting design specifications of a second order system using Matlab
Therefore, it is not possible to design the system by just varying the
value of K

I think that the mistake you are making is assuming \$\omega_n\$ is unaffected by \$K\$. In your closed-loop \$G(s)\$ formula, \$\omega_n\$ is in fact this: -
$$\sqrt{23.8 + 1.247\cdot K}$$
Hence, \$\zeta = \dfrac{5.02}{\sqrt{23.8 + 1.247\cdot K}}\$