homework and exercises - How can I derive the Hamiltonian of simple harmonic oscillator from this Lagrangian?

I'm working through Leonard Susskind's Theoretical Minimum: Classical Mechanics and I can't seem to understand how the Hamiltonian of a simple harmonic oscillator is derived from the following Lagrangian:

$$L~=~\frac{\omega}{2} \dot q^2 - \frac{\omega}{2}q^2$$

where $\omega=\sqrt{\frac{k}{m}}$.

The main problem I'm having is that while I understand the basic substitution required to transform the Lagrangian into the Hamiltonian form, I can't understand how the final form of the Hamiltonian is derived in the book:

$$H~=~\frac{\omega}{2}(p^2 + q^2) $$

I made it this far:

$$H= \frac{p^2}{m} - \frac{p^2}{2m^2\omega} + \omega\frac{q^2}{2}$$

I don't see the intuition behind the substitutions necessary to derive the final form of this equation. How does $\frac{p^2}{m} - \frac{p^2}{2m^2\omega}$ become $\frac{\omega}{2}p^2$?