Finding the natural frequency
How do you find the value of natural frequency without knowing the tau constants? Is it even possible?
  


No.
The problem is that there is 1 equation with 2 unknowns. You need to know either \$\tau_1\$ or \$\tau_2\$ as shown below.

Substituting \$\zeta=0.5 \$ in
$$ 2\zeta\omega_n = \omega_n = \frac{1+100\tau_2}{\tau_1} $$

and using 
$$\omega_n^2 = \frac{100}{\tau_1}$$

gives 1 equation with 2 unknowns:

$$ \sqrt { \frac{100}{\tau_1} } = \frac{1+100\tau_2}{\tau_1} $$
or
$$10 \sqrt { \tau_1 } = 1+100\tau_2 $$

If you know one of the unknowns, you can find rest using:

$$ \tau_1 = \frac{ (1+100\tau_2 )^2}{100} $$

$$ \tau_2 = \frac{ 10\sqrt{ \tau_1 } - 1 }{100} $$

and
$$\omega_n = \sqrt { \frac{100}{\tau_1} } =  \frac{100}{1+100\tau_2} $$