Is it possible to find the voltage gain of the constant gain block of this control system?
The procedure is quite simple.
When you have the BODE plot for the LOOP GAIN (open-loop transfer function) you only must shift the magnitude plot up or down until the desired phase margin is met. This shifting will not influence the phase plot. As a result of this shifting the frequency where the magnitude crosses the horizontal axis (0 dB) will be shifted as well.
Of course, the amount of shifting gives you the required constant gain (in dB).
Remember that the loop gain is nothing else than the product of the remaining blocks (better; the corresponding transfer function). In the BODE plot, both functions are to be added, of course (due to the log display).
Are you familiar with the definition of the phase margin? It is defined NOT for the closed-loop but for the loop gain!