This $E = \frac{1}{2}mv^2$ is only correct in the limit of $v \ll c$ (the Newtonian or non-relativistic limit). The generally correct expression is $$E = \sqrt{(mc^2)^2 + (pc)^2}\,.$$ It follows that rest of your work is also incorrect.

Never mix relativistically correct and Newtonian math. Unless you really know what you are doing, of course.