Drawing on results of Derdzi\'nski's from the 80's, we classify conformally K\"ahler, $U(2)$-invariant, Einstein metrics on the total space of $\mathcal{O}(-m)$, for all $m \in \mathbb{N}$. This yields infinitely many $1$-parameter families of metrics exhibiting several different behaviours including asymptotically hyperbolic metrics (more specifically of Poincar\'e type), ALF metrics, and metrics which compactify to a Hirzebruch surface $\mathbb{H}_m$ with a cone singularity along the ''divisor at infinity''. This allows us to investigate transitions between behaviours yielding interesting results. For instance, the recently discovered Ricci-flat ALF metric known as the Chen-Teo instanton can be obtained as a limit of a family of cone angle Einstein metrics on $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$ when the cone angle converges to zero. We also construct Einstein metrics which are asymptotically hyperbolic and conformal to a scalar-flat K\"ahler metric. Such metrics cannot be obtained by applying Derdzi\'nski's theorem.