Transfer function and cut-off frequency of cascoded RC-LRC filter
You need to solve:

$$\frac{1}{\sqrt{2}}\cdot\left|\underline{\mathcal{H}}\left(\hat{\omega}\right)\right|=\left|\underline{\mathcal{H}}\left(\omega\right)\right|\space\Longleftrightarrow\space\omega=\dots\tag1$$

Using Mathematica I got (assuming that your transfer function is correct) for \$\hat{\omega}\$:



And \$\left|\underline{\mathcal{H}}\left(\hat{\omega}\right)\right|\$ gives:



So, solving equation \$(1)\$ gives:



Which gives: