I think there should be $-i$ in the expression for $\beta$:

$$R_x\left(\frac{\pi}{4}\right) |0\rangle= \begin{pmatrix} \cos\left(\frac{\pi}{8}\right) & -i \sin\left(\frac{\pi}{8}\right) \\ -i \sin\left(\frac{\pi}{8}\right) & \cos\left(\frac{\pi}{8}\right) \end{pmatrix} \begin{pmatrix} 1 \\ 0\end{pmatrix} = \frac{\sqrt{2 + \sqrt{2}}}{2} |0\rangle - i \frac{\sqrt{2 - \sqrt{2}}}{2} |1\rangle$$

So, we will have:

$$|\psi\rangle = \frac{\sqrt{2 + \sqrt{2}} -i \sqrt{2 - \sqrt{2}}}{2\sqrt{2}} |+\rangle + \frac{\sqrt{2 + \sqrt{2}} + i \sqrt{2 - \sqrt{2}}}{2\sqrt{2}} |-\rangle$$

Then:

$$\left|\frac{\sqrt{2 + \sqrt{2}} -i \sqrt{2 - \sqrt{2}}}{2\sqrt{2}}\right|^2 = 0.5 $$

I think there should be $-i$ in the expression for $\beta$:

\begin{equation} R_x\left(\frac{\pi}{4}\right) |0\rangle= \begin{pmatrix} \cos\left(\frac{\pi}{8}\right) & -i \sin\left(\frac{\pi}{8}\right) \\ -i \sin\left(\frac{\pi}{8}\right) & \cos\left(\frac{\pi}{8}\right) \end{pmatrix} \begin{pmatrix} 1 \\ 0\end{pmatrix} = \\ =\frac{\sqrt{2 + \sqrt{2}}}{2} |0\rangle - i \frac{\sqrt{2 - \sqrt{2}}}{2} |1\rangle \end{equation}

So, we will have:

$$|\psi\rangle = \frac{\sqrt{2 + \sqrt{2}} -i \sqrt{2 - \sqrt{2}}}{2\sqrt{2}} |+\rangle + \frac{\sqrt{2 + \sqrt{2}} + i \sqrt{2 - \sqrt{2}}}{2\sqrt{2}} |-\rangle$$

Then:

$$\left|\frac{\sqrt{2 + \sqrt{2}} -i \sqrt{2 - \sqrt{2}}}{2\sqrt{2}}\right|^2 = 0.5 $$