Because $MG$ and $MH$ are tangents to $k$ we have that $OG$ is perpendicular to $MG$ and $OH$ is perpendicular to $MH$ so $MGOH$ is cyclic. Hence ${\angle}MGH = {\angle}MOH$ and ${\angle}MOH = {\angle}MOG$ because of symmetry. Also $GK$ is perpendicular to $MH$, and because $MO$ is perpendicular to $GH$, and $OH$ to is perpendicualr to $MH$, we have that ${\angle}OKG = {\angle}KOH = {\angle}MOH = {\angle}MHG$, hence ${\angle}OGK = {\angle}GMH$.

$OG\bot GM, GK\bot MH\implies\angle OGK=\angle GMH$, as having perpendicluar sides. Best regards, sunken rock