You want the choose the largest value of the function $\frac{1}{|z^2 + 2|}$ on the line from $0$ to $1+i$. One useful way to view these problems is to geometrically interpret them.
The image of the line $\gamma$ under the function $z^2$ is given by the vertical line from $0$ to $2$. How do I know this? The line $\gamma$ forms angle $\frac{\pi}{4}$ with the real axis. The function $z^2$ doubles the angle as measured from the real axis so that $\gamma$ is taken to a line with angle $\frac{\pi}{2}$, i.e. the vertical line. The function also squares magnitudes so that the endpoint with original magnitude $\sqrt{2}$ now has magnitude $\sqrt{2}^2 = 2$. It is now easy to see that the value of $|z^2 + 2|$ will be minimized when $z=0$ so that the maximum of $\frac{1}{|z^2 + 2|}$ on $\gamma$ is precisely $\frac{1}{2}$.