A similar theorem is proved in <a href="http://www.math.unl.edu/~shartke2/math/papers/min-deg-k-factor.pdf">"Relating minimum degree and the existence of a k-factor"</a> by Hartke, Martin and Seacrest. 

They show that a graph $G$ on $n$ vertices with minimum degree $\delta\geq \frac{n}{2}$ contains a $k$-factor if $kn$ is even and $$k< \frac{\delta+\sqrt{2\delta n-n^2+8}}{2}.$$

Moreover they show that this is optimal up to a small additive constant ($\le 1$). Notice that as $\delta\to n$ we have $k\to n$, as well.