We have that $ \mathbf{D} = \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} $ and $ \mathbf{R} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}, $ so \[\mathbf{D} \mathbf{R} = \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} = \begin{pmatrix} k \cos \theta & -k \sin \theta \\ k \sin \theta & k \cos \theta \end{pmatrix}.\]Thus, $ k \cos \theta = -7 $ and $ k \sin \theta = -1 $. Then \[k^2 \cos^2 \theta + k^2 \sin^2 \theta = 49 + 1 = 50,\]which simplifies to $ k^2 = 50 $. Since $ k > 0, $ $ k = \sqrt{50} = \boxed{5 \sqrt{2}} $.