How do I get a faster system response?
The transfer function is $H(s) = \frac{16.94s + 579.5}{s^2 + 507.2s + 1224}$
This transfer function has 2 poles, one slow pole at -2.4248 and a fast pole at -504.7752. The function has a slowish zero at -34.2. Good  news, your poles and zero are all in the left-half plane. It is much easier to control a system with zeroes and poles in the left-half plane than in the right-half plane. Your slow pole at -2.4248 will limit your performance. Your slowish zero at -34.2 will probably limit your performance by creating some overshoot, or maybe not. Your fast pole will have a limited impact on the closed-loop performance, so you can simply ignore it for now.
You could design your controller to cancel the slow pole at -2.4248 and the slowish zero at -34.2. This would be a lead lag compensator looking like this
$$H_{LeadLag} = K\frac{s + 2.4248}{s+34.2}$$
Then you could set a PI in series with the lead-lag compensator and tune the P and I gain to get the closed-loop performance you want.
$$H_{c} = H_{LeadLag} \frac{K_ps + K_i}{s}$$
That being said, I'm not sure my solution is particularly robust. You should check the Gang of four stability functions https://www.cds.caltech.edu/~murray/courses/cds101/fa02/caltech/astrom-ch5.pdf.
Edit 2 :
Another solution, simply use a PI controller and set the ratio of the P and I gain to 2.4248 to cancel your slow pole. This controller will not cancel your slowish zero, but maybe you don't need to cancel it to get the performance you want.
$$H_{PI} = K\frac{s+2.4248}{s}$$