What is wrong with my proof by contradiction that if $n^2$ is divisible by 3, then $n^2$ is divisible by 9?

"Assume that $n^2 = 3k$ ($k$ being any non-zero integer)."

There is the problem. $k$ is some integer. Since $n^2$ is a fixed number, $k$ cannot be anything you want.

The place where you attempt to reach a contraction, "But we have previously said that $k$ can be any non-zero integer" therefore doesn't work.