jmc

Writing the equation of the ellipse in the form $$\frac{x^2}{(1/\sqrt{k}{)}^2}+\frac{y^2}{1^2}=1,$$we see that the lengths of the semi-horizontal and semi-vertical axis are $\frac{1}{\sqrt{k}}$ and $1,$ respectively. Since $k>1,$ the vertical axis is the longer (major) axis. Then the distance from the center of the ellipse, the origin, to each focus is $$\sqrt{1-{\left(\sqrt{\frac{1}{k}}\right)}^2}=\frac{\sqrt{k-1}}{\sqrt{k}}.$$ The existence of such a circle implies that the origin is equidistant from each focus and each endpoint of the horizontal (minor) axis. Therefore, we have $$\frac{\sqrt{k-1}}{\sqrt{k}}=\frac{1}{\sqrt{k}},$$so $\sqrt{k-1}=1.$ Thus, $k-1=1,$ and $k=\boxed{2}.$