Rather than tell you what is wrong with your method, I prefer to tell you how I think you should approach the problem. The derivative of V with respect to theta in degrees, using the principles of calculus is:
dV/dtheta = (0.5*k*L^2*2*sind(theta)*cosd(theta) - 0.5*m*g*L*sind(theta))*pi/180
= 0.5*L*pi/180*sind(theta) * (2*k*L*cosd(theta)-m*g)
Expressed in this factored form it is quickly evident that the derivative is zero whenever sind(theta) = 0 or when cosd(theta) = m*g/(2*k*L). Hence its roots are:
theta = 180*n
for any integer n, along with
theta = acosd(m*g/(2*k*L)) or 360-acosd(m*g/(2*k*L))
together with any integral multiple of 360 degrees added or subtracted from these. Using that method is very much more satisfactory than resorting to an iterative approach. (By the way, it's a lot easier to deal with derivatives of the trigonometric functions if you use radians rather than degrees.)
