Just because $y=2$, it does not mean that $x =0$. In particular, you want that, for any $x$, $y=2$, but $x$ is not $0$. Since you already know that
and that to change to polar coordinates, you can use
$$y= \rho \sin \theta$$,
we plug in $y=2$, which gives
$$2= \rho \sin \theta$$
$$2 \csc \theta= \rho $$
Note that we don't consider the variable $x$ and $$x= \rho \cos \theta$$, since it suffices to use only the $y$-equation above, because it fully describes the dependence between $\theta$ and $\rho$.
In particular note the radius can't be always $2$, else you would get a circle! Everytime you state something such as $x=0$ or $r=2$, think the implications it carries, and it might help you to spot the flaw.