Example 5 As shown in Figure 5, in $\triangle A B C$, $A C=7, B C=$ $4, D$ is the midpoint of $A B$, $E$ is a point on side $A C$, and $\angle A E D=$ $90^{\circ}+\frac{1}{2} \angle C$. Find the length of $C E$.
As shown in Figure 5, draw $B F / / D E$ intersecting $A C$ at point $F$, and draw the angle bisector of $\angle A C B$ intersecting $A B$ at point $G$, and intersecting $B F$ at point $H$. Then $\angle A E D=\angle A F B=\angle C H F+\frac{1}{2} \angle C$.
From $\angle A E D=90^{\circ}+\frac{1}{2} \angle C$, we get
$$
\angle C H F=90^{\circ}=\angle C H B \text {. }
$$

Since $\angle F C H=\angle B C H$, then $C F=C B=4$, which means $A F=A C-C F=7-4=3$.
It is easy to see that $A E=E F=1.5$.
Therefore, $C E=5.5$.