The sides of a triangle have lengths $11$,$15$, and $k$, where $k$ is an integer. For how many values of $k$ is the triangle obtuse?

$\textbf{(A)}\ 5 \qquad
\textbf{(B)}\ 7 \qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 13 \qquad
\textbf{(E)}\ 14$
1. **Determine the range of \( k \) using the triangle inequality:**
   The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Therefore, we have:
   \[
   11 + 15 > k \implies k < 26
   \]
   \[
   11 + k > 15 \implies k > 4
   \]
   \[
   15 + k > 11 \implies k > -4 \quad (\text{This is always true since } k > 4)
   \]
   Combining these inequalities, we get:
   \[
   4 < k < 26
   \]

2. **Use the Law of Cosines to determine when the triangle is obtuse:**
   A triangle is obtuse if one of its angles is greater than \( 90^\circ \). For an angle \( \theta \) to be obtuse, \( \cos(\theta) \) must be negative. The Law of Cosines states:
   \[
   c^2 = a^2 + b^2 - 2ab \cos(\theta)
   \]
   We will check each angle to see when \( \cos(\theta) \) is negative.

3. **Check the angle opposite the side of length 11:**
   \[
   11^2 = 15^2 + k^2 - 2 \cdot 15 \cdot k \cdot \cos(\theta)
   \]
   \[
   121 = 225 + k^2 - 30k \cos(\theta)
   \]
   \[
   -104 = k^2 - 30k \cos(\theta)
   \]
   \[
   \cos(\theta) = \frac{k^2 + 104}{30k}
   \]
   Since \( k > 4 \), \( k^2 + 104 \) is always positive, and \( 30k \) is always positive, so \( \cos(\theta) \) is always positive. Therefore, the angle opposite the side of length 11 is always acute.

4. **Check the angle opposite the side of length 15:**
   \[
   15^2 = 11^2 + k^2 - 2 \cdot 11 \cdot k \cdot \cos(\theta)
   \]
   \[
   225 = 121 + k^2 - 22k \cos(\theta)
   \]
   \[
   104 = k^2 - 22k \cos(\theta)
   \]
   \[
   \cos(\theta) = \frac{k^2 - 104}{22k}
   \]
   For \( \cos(\theta) \) to be negative:
   \[
   \frac{k^2 - 104}{22k} < 0
   \]
   \[
   k^2 - 104 < 0
   \]
   \[
   k^2 < 104
   \]
   \[
   k < \sqrt{104} \approx 10.2
   \]
   Since \( 4 < k < 10.2 \), the integer values of \( k \) are \( 5, 6, 7, 8, 9, 10 \). There are 6 values that satisfy this condition.

5. **Check the angle opposite the side of length \( k \):**
   \[
   k^2 = 11^2 + 15^2 - 2 \cdot 11 \cdot 15 \cdot \cos(\theta)
   \]
   \[
   k^2 = 121 + 225 - 330 \cos(\theta)
   \]
   \[
   k^2 = 346 - 330 \cos(\theta)
   \]
   \[
   \cos(\theta) = \frac{346 - k^2}{330}
   \]
   For \( \cos(\theta) \) to be negative:
   \[
   \frac{346 - k^2}{330} < 0
   \]
   \[
   346 - k^2 < 0
   \]
   \[
   k^2 > 346
   \]
   \[
   k > \sqrt{346} \approx 18.6
   \]
   Since \( 18.6 < k < 26 \), the integer values of \( k \) are \( 19, 20, 21, 22, 23, 24, 25 \). There are 7 values that satisfy this condition.

6. **Combine the results:**
   The total number of values of \( k \) that make the triangle obtuse is:
   \[
   6 + 7 = 13
   \]

The final answer is \(\boxed{13}\)