AMC 10 2017b Problem 21

In $\triangle ABC$, $AB=6$, $AC=8$, $BC=10$, and $D$ is the midpoint of $\overline{BC}$. What is the sum of the radii of the circles inscribed in $\triangle ADB$ and $\triangle ADC$? $\textbf{(A)}\ \sqrt{5} \textbf{(B)}\ \frac{11}{4} \textbf{(C)}\ 2\sqrt{2} \textbf{(D)}\ \frac{17}{6} \textbf{(E)}\ 3$