II. (25 points) Given that $D$ is a point inside $\triangle A B C$, $E$ is the midpoint of side $A C$, $A B=6, B C=10, \angle B A D=$ $\angle B C D, \angle E D C=\angle A B D$. Find the length of $D E$.

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II. As shown in Figure 4, extend \( CD \) to point \( F \) such that \( DF = CD \), and connect \( AF \) and \( BF \).
Then \( AF \parallel DE \), and \( DE = \frac{1}{2} AF \).
Thus, \( \angle AFD = \angle EDC = \angle ABD \)
\(\Rightarrow A, F, B, D\) are concyclic
\(\Rightarrow \angle BFD = \angle BAD = \angle BCD \)
\(\Rightarrow BF = BC = 10 \).
Since \( D \) is the midpoint of \( FC \), it follows that \( BD \perp FC \).
Therefore, \( \angle FAB = \angle FDB = 90^\circ \).
In the right triangle \( \triangle FAB \),
\( FA = \sqrt{BF^2 - AB^2} = \sqrt{10^2 - 6^2} = 8 \).
Hence, \( DE = \frac{1}{2} FA = 4 \).