Dissipation and Impedance
You have an error in your calculation. $$Z = \frac{\omega^2 L^2 R}{\omega^2 L^2 + R^2} + j\frac{\omega L R^2}{\omega^2 L^2 + R^2}$$

The easy way to do this is to note that the admittance is easily calculated as $$Y = \frac{1}{R} + \frac{1}{j\omega L} = \frac{1}{R}  - j\frac{1}{\omega L}$$

So the current is $$i = Yv = \frac{v}{R} - j\frac{v}{\omega L}$$.

The power is then $$P = i\,v = \frac{v^2}{R} - j\frac{v^2}{\omega L}$$

Note that the real part only depends on the resistance, not the inductance.  You can take your (corrected) value for the impedance and use it to calculate the current, then go through a bunch of gyrations of complex-number algebra to come up with my expression for current, and then come to the same conclusion.