Singular Values & the SVD · Challenge problem

Let $P$ be the (oblique) projection of $\mathbb R^2$ onto the line spanned by $(1,1)$ along the line spanned by $(1,-2)$: $P(1,1) = (1,1)$ and $P(1,-2) = 0$. Find the spectral norm $\|P\|_2$. Give your answer in simplest radical form.