Confusion regarding finding closed loop transfer function and damping constant from open-loop transfer function
As user215805 stated, the standard form is
$$TF=K\frac{\omega^2_n}{s^2+2\delta\omega_ns+\omega^2_n}$$
And you have
$$TF=\frac{1.247}{s^2+9.76s+25.047}$$
This means that in your case $$\omega^2_n=25.047$$
Which you already know, but notice that from the standard form you can deduce that
$$K\omega^2_n=1.247$$
And simply
$$K=\frac{1.247}{\omega^2_n}=\frac{1.247}{25.047}≈0.05$$
So now you have
$$TF=0.05\frac{25.047}{s^2+9.76s+25.047}$$
Similarly you can deduce that
$$2\delta\omega_n=9.76$$
And therefore
$$\delta=\frac{9.76}{2\omega_n}≈0.975$$
And so you have reached the standard form
$$TF=0.05\frac{25.047}{s^2+2(0.975)\sqrt{25.047}s+25.047}$$