Phase margin for a system (design of PD controller)
If I understood well, you want to compensate the plant whose transfer function is given so that once compensated you have a phase margin of 35°. The plant transfer function is \$H(s)=H_0\frac{1}{\tau_2s^2+\tau_1s+1}\$ with \$H_0=\frac{1}{k_f}\$, \$\tau_1=\frac{1}{20k_f}s\$ and \$\tau_2=\frac{1}{20k_f}s^2\$. The plot I got from Mathcad is this one and matches yours:



Then, you need to pick a crossover frequency \$f_c\$ at which you will measure the open-loop phase margin of 35°. Let's take \$f_c=10\;Hz\$. What is the attenuation of the plant at \$f_c\$ and what is its phase?



From these values, calculate the needed phase boost from the inverting compensator to meet the 35° design criteria and deduce the position of the zero:



Determine what the \$k_p\$ coefficient should be based on the zero position and plot the open-loop gain \$T(s)\$:



The crossover frequency is 10 Hz as arbitrarily selected and the phase margin is 35°. The compensator transfer function is \$G_1(s)\$ in the picture. Is this what you wanted to see?