To find the eccentricity of the hyperbola, we need to use the given information about the length of the conjugate axis and the distance between the foci.

The formula for the eccentricity of a hyperbola is given by:

e = c/a

Where 'e' is the eccentricity, 'c' is the distance between the foci, and 'a' is the distance from the center to a vertex.

Let's break down the problem into steps to find the eccentricity.

Step 1: Finding the length of the transverse axis
The length of the conjugate axis is given as 5. The length of the transverse axis is twice the length of the conjugate axis. Therefore, the length of the transverse axis is 2 * 5 = 10.

Step 2: Finding the distance from the center to a vertex
The distance from the center to a vertex is half the length of the transverse axis. Therefore, the distance from the center to a vertex is 10 / 2 = 5.

Step 3: Finding the distance from the center to a focus
The distance from the center to a focus is given by the equation:

c^2 = a^2 + b^2

Where 'c' is the distance between the foci, 'a' is the distance from the center to a vertex, and 'b' is the distance from the center to a co-vertex.

Since the length of the conjugate axis is 5, the distance from the center to a co-vertex is 5. Therefore, we can rewrite the equation as:

13^2 = 5^2 + b^2

169 = 25 + b^2
b^2 = 169 - 25
b^2 = 144
b = √144
b = 12

Step 4: Finding the eccentricity
Now that we have the distance from the center to a vertex (a = 5) and the distance between the foci (c = 13), we can calculate the eccentricity using the formula:

e = c/a
e = 13/5
e = 13/12

Therefore, the eccentricity of the hyperbola is 13/12, which corresponds to option 'D'.