Prove that $\alpha:=\sqrt{\pi}+\sqrt 2 \in \Bbb{C} $ is trancendental over $\Bbb{Q}$.

If it was algebraic then, since the algebraic numbers form a field and $-\sqrt2$ is algebraic,$$\left(\sqrt\pi+\sqrt2\right)-\sqrt2$$would be algebraic too. But this number is $\sqrt\pi$, which is transcendental, since $\pi$ is transcendental.

You must use the fact that $\pi$ (and so $\sqrt{\pi}$) is transcendental.

Before proving your proposition, I would like to note the fact that if $R$ is a subring of some ring $P$ then the integral elements form a ring (for a proof, go on my post: Proof that the algebraic integers form a subring of $\mathbb{C}$ and replace $\mathbb{Z}$ with $R$ and $\mathbb{C}$ with $P$).

Consider now the evaluation homomorphism $\psi_{\alpha}: \mathbb{Q}[T] \longmapsto \mathbb{Q}(\sqrt{\pi}; \sqrt{2})$, then \begin{equation} \frac{\mathbb{Q}[T]}{<P_{\alpha}>} \cong \mathbb{Q}(\alpha) \end{equation}, where $P_{\alpha}$ is the minimal polynominal of $\alpha$.

From my proof that on the link above you can notice that \begin{equation}dim_{\mathbb{Q}}(\mathbb{Q}(\alpha)) \leq dim_{\mathbb{Q}}(\mathbb{Q}(\sqrt{\pi};\sqrt{2})) \leq dim_{\mathbb{Q}}(\mathbb{Q}(\sqrt{\pi}) \otimes_{\mathbb{Q}}\mathbb{Q}(\sqrt{2}))\end{equation} but since \begin{equation} \mathbb{Q}(\sqrt{\pi};\sqrt{2})= \mathbb{Q}(\sqrt{\pi})\mathbb{Q}(\sqrt{2}) \end{equation}, where by some fields $K$ and $F$, by $KF$ we mean the composite extension also, \begin{equation} dim_{\mathbb{Q}}(\mathbb{Q}(\sqrt{\pi}) \otimes_{\mathbb{Q}}\mathbb{Q}(\sqrt{2}))= dim_{\mathbb{Q}}(\mathbb{Q}(\sqrt{\pi})\mathbb{Q}(\sqrt{2})) \Rightarrow dim_{\mathbb{Q}}(\mathbb{Q}(\sqrt{\pi}) \otimes_{\mathbb{Q}}\mathbb{Q}(\sqrt{2}))=dim_{\mathbb{Q}}(\mathbb{Q}(\sqrt{\pi};\sqrt{2})) \end{equation}.
Since $\sqrt{{\pi}}$ is transcendental, \begin{equation}dim_{\mathbb{Q}}(\mathbb{Q}(\sqrt{\pi};\sqrt{2}))= \infty \end{equation}.
Since $\mathbb{Q}(\sqrt{\pi}) \subseteq \mathbb{Q}(\alpha) \subseteq \mathbb{Q}(\sqrt{\pi};\sqrt{2})$ and $dim_{\mathbb{Q}}(\mathbb{Q}(\sqrt{\pi}))= \infty$ (because $\sqrt{\pi}$ is transcendental), obviously: \begin{equation} dim_{\mathbb{Q}}(\mathbb{Q}(\alpha))= \infty \end{equation} and hence $P_{\alpha}=0$ and hence $\alpha$ is transcendental. $\,\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \square$

Continuing your argument: $$ \pi \in \Bbb{Q}(\sqrt \pi) \subseteq \Bbb{Q}(\sqrt \pi, \sqrt 2) = \Bbb{Q}(\alpha) \implies \Bbb{Q}(\pi) \subseteq \Bbb{Q}(\alpha) \implies [\Bbb{Q}(\alpha ):\Bbb{Q}] \ge [\Bbb{Q}(\pi):\Bbb{Q}]=\infty $$