A sequence $(x_k)$ converges to 0 superlinearly if $||x_{k+1}/x_k||\to0$ as $k\to\infty$.
A sequence $(x_k)$ converges to 0 quadratically if there exist $N,M>0$ such that $||x_{k+1}/x_k^2||<M$ for all $k>N$.
So far, to prove superlinearity, I have that $||\frac{k^k}{(k+1)^{(k+1)}}|| = ||(\frac{k}{k+1})^k\frac{1}{k+1}||\to 0$ as $k\to\infty$ since $\frac{k}{k+1}<1$ for all $k$. I feel like that argument is not sufficient but have not been able to come up with anything better. Edit: a very helpful comment has helped me tighten this argument.
I have also attempted to show that assuming there exists $N,M>0$ such that $||\frac{k^{2k}}{(k+1)^{(k+1)}}||<M$ for all $k>N$ leads to a contradiction. Lets pick $k=N+1$. Then we have $||\frac{(N+1)^{2N+2}}{(N+2)^{(N+2)}}||<M$. So if we can show that there is no M such that this holds, we are done.
Any guidance would be appreciated!