A park is in the shape of a regular hexagon $2$ km on a side.  Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km.  How many kilometers is she from her starting point?

$ \textbf{(A)}\ \sqrt{13}\qquad\textbf{(B)}\ \sqrt{14}\qquad\textbf{(C)}\ \sqrt{15}\qquad\textbf{(D)}\ \sqrt{16}\qquad\textbf{(E)}\ \sqrt{17}$
1. **Understanding the problem**:
   - The park is a regular hexagon with each side measuring 2 km.
   - Alice starts at one corner and walks 5 km along the perimeter.
   - We need to determine her distance from the starting point after walking 5 km.

2. **Hexagon properties**:
   - A regular hexagon can be divided into 6 equilateral triangles.
   - Each side of the hexagon is 2 km.

3. **Walking along the perimeter**:
   - Alice walks 5 km along the perimeter.
   - Since each side is 2 km, she will cover 2 sides completely (4 km) and then walk an additional 1 km on the third side.

4. **Position after walking 5 km**:
   - Let's place the hexagon on a coordinate plane for simplicity.
   - Assume Alice starts at vertex \( A \) at coordinates \( (0, 0) \).
   - The vertices of the hexagon can be labeled as follows:
     - \( A = (0, 0) \)
     - \( B = (2, 0) \)
     - \( C = (3, \sqrt{3}) \)
     - \( D = (2, 2\sqrt{3}) \)
     - \( E = (0, 2\sqrt{3}) \)
     - \( F = (-1, \sqrt{3}) \)

5. **Walking path**:
   - Alice walks from \( A \) to \( B \) (2 km).
   - Then from \( B \) to \( C \) (2 km).
   - She has walked 4 km so far and needs to walk 1 more km.
   - From \( C \) to \( D \) is another side, but she only walks 1 km along this side.

6. **Finding the coordinates after 5 km**:
   - The direction from \( C \) to \( D \) is along the line segment connecting these points.
   - The coordinates of \( C \) are \( (3, \sqrt{3}) \).
   - The coordinates of \( D \) are \( (2, 2\sqrt{3}) \).
   - The direction vector from \( C \) to \( D \) is \( (2 - 3, 2\sqrt{3} - \sqrt{3}) = (-1, \sqrt{3}) \).
   - Normalize this vector to get the unit direction vector: \( \left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right) \).

7. **Walking 1 km along this direction**:
   - Starting from \( C \) and moving 1 km in the direction \( \left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right) \):
   - New coordinates: \( \left( 3 - \frac{1}{2}, \sqrt{3} + \frac{\sqrt{3}}{2} \right) = \left( \frac{5}{2}, \frac{3\sqrt{3}}{2} \right) \).

8. **Distance from the starting point**:
   - The starting point is \( (0, 0) \).
   - The ending point is \( \left( \frac{5}{2}, \frac{3\sqrt{3}}{2} \right) \).
   - Use the distance formula:
     \[
     d = \sqrt{\left( \frac{5}{2} \right)^2 + \left( \frac{3\sqrt{3}}{2} \right)^2 }
     \]
     \[
     d = \sqrt{\frac{25}{4} + \frac{27}{4}}
     \]
     \[
     d = \sqrt{\frac{52}{4}}
     \]
     \[
     d = \sqrt{13}
     \]

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