By the mean value theorem,
$$\begin{align*}
f(x_0 + h, y_0 + k) - f(x_0, y_0)
&= f(x_0 + h, y_0 + k) - f(x_0, y_0 + k) + f(x_0, y_0 + k) - f(x_0, y_0)\\
& = hD_x f(x_0 + \xi, y_0 + k) + f(x_0, y_0 + k) - f(x_0, y_0)
\end{align*}$$
(For some $\xi$ between $0$ and $h$.)
Hence:
$$\begin{align*}
&\frac{f(x_0 + h, y_0 + k) - f(x_0, y_0) - hD_x f(x_0, y_0) - k D_y f(x_0, y_0)}{\sqrt{h^2 + k^2}}\\
&= \frac{h}{\sqrt{h^2 + k^2}}(D_x f(x_0 + \xi, y_0 + k) - D_x f(x_0, y_0))\\
&\quad + \frac{k}{\sqrt{k^2 + h^2}}\left(\frac{f(x_0, y_0 + k) - f(x_0, y_0)}{k} - D_y f(x_0, y_0) \right)
\end{align*}$$
If you take the limit as $(h, k) \to 0$, the remaining fractions are bounded, the first bracket vanishes by continuity, and the second by definition. QED