Let $D$ and $E$ be points on side $[BC]$ of a triangle $ABC$ with circumcenter $O$ such that $D$ is between $B$ and $E$, $|AD|=|DB|=6$, and $|AE|=|EC|=8$. If $I$ is the incenter of triangle $ADE$ and $|AI|=5$, then what is $|IO|$?

$ 
\textbf{(A)}\ \dfrac {29}{5}
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ \dfrac {23}{5}
\qquad\textbf{(D)}\ \dfrac {21}{5}
\qquad\textbf{(E)}\ \text{None of above}
$
1. **Identify the given information and setup the problem:**
   - Points \( D \) and \( E \) are on side \( [BC] \) of triangle \( ABC \).
   - \( D \) is between \( B \) and \( E \).
   - \( |AD| = |DB| = 6 \) and \( |AE| = |EC| = 8 \).
   - \( I \) is the incenter of triangle \( ADE \) and \( |AI| = 5 \).
   - We need to find \( |IO| \), where \( O \) is the circumcenter of \( \triangle ABC \).

2. **Determine the properties of the triangle and the points:**
   - Since \( D \) and \( E \) are on \( [BC] \), \( D \) and \( E \) divide \( BC \) into segments \( BD = 6 \), \( DE \), and \( EC = 8 \).
   - The perpendicular bisector of \( AB \) passes through \( O \) and \( D \), making \( OD \) an external angle bisector of \( \angle ADE \).
   - Similarly, the perpendicular bisector of \( AC \) passes through \( O \) and \( E \), making \( OE \) an external angle bisector of \( \angle AED \).

3. **Establish the relationship between \( O \) and \( \triangle ADE \):**
   - Since \( O \) is the circumcenter of \( \triangle ABC \), it lies on the perpendicular bisectors of \( AB \) and \( AC \).
   - Therefore, \( O \) is an excenter of \( \triangle ADE \), specifically the excenter opposite \( A \).

4. **Use the similarity of triangles to find \( AO \):**
   - Since \( A, I, O \) are collinear, we can use the similarity of triangles \( \triangle DAI \) and \( \triangle OAE \).
   - The angles \( \angle ADI \) and \( \angle AOE \) are equal, and \( \angle DAI \) and \( \angle OAE \) are equal.
   - Therefore, \( \triangle DAI \sim \triangle OAE \).

5. **Set up the proportion and solve for \( AO \):**
   - From the similarity, we have:
     \[
     \frac{AD}{AO} = \frac{AI}{AE}
     \]
   - Substituting the known values:
     \[
     \frac{6}{AO} = \frac{5}{8}
     \]
   - Solving for \( AO \):
     \[
     AO = \frac{6 \times 8}{5} = \frac{48}{5}
     \]

6. **Find \( IO \) using the relationship between \( A, I, \) and \( O \):**
   - Since \( A, I, O \) are collinear and \( AI = 5 \), we can find \( IO \) by subtracting \( AI \) from \( AO \):
     \[
     IO = AO - AI = \frac{48}{5} - 5 = \frac{48}{5} - \frac{25}{5} = \frac{23}{5}
     \]

The final answer is \( \boxed{\frac{23}{5}} \).