Affine Beilinson-Bernstein localization at the critical level for $G{L}_2$

We prove the rank 1 case of a conjecture of Frenkel-Gaitsgory: critical level Kac-Moody representations with regular central characters localize onto the affine Grassmannian. The method uses an analogue in local geometric Langlands of the existence of Whittaker models for most representations of $G{L}_2$ over a non-Archimedean field.