jmc

We want to have that $f(f(x))=x$ for every $x.$ If $x=0$ then $f(f(0))=f(0)=0,$ so we are fine.

Since $f$ applied to any negative number returns a positive number, and we can get all positive numbers this way, applying $f$ to any positive number must give a negative number. Therefore $k(x)<0$ for any $x>0.$

If $x>0$ and $f$ is its own inverse then $$x=f(f(x))=f(k(x))=-\frac{1}{2k(x)},$$where in the last step we used that $k(x)<0.$

Solving this for $k$ gives $$k(x)=\boxed{-\frac{1}{2x}}.$$