First, integrate the function \(\frac{1}{k} x^{\frac{1}{2}} + x^{-\frac{1}{2}} + \frac{1}{k^2}\):

\(\int \left( \frac{1}{k} x^{\frac{1}{2}} + x^{-\frac{1}{2}} + \frac{1}{k^2} \right) \, dx = \left[ \frac{2x^{\frac{3}{2}}}{3k} \right] + \left[ 2x^{\frac{1}{2}} \right] + \left[ \frac{x}{k^2} \right]\)

Apply the limits \(\frac{1}{4}k^2 \to k^2\) to the integrated expression:

\(\left( \frac{2k^2}{3} + 2k + 1 \right) - \left( \frac{k^2}{12} + k + \frac{1}{4} \right)\)

This simplifies to:

\(\frac{7}{12} k^2 + k + \frac{3}{4}\)

Equate this to \(\frac{13}{12}\) and simplify to a quadratic equation:

\(\frac{7}{12} k^2 + k + \frac{3}{4} = \frac{13}{12}\)

\(7k^2 + 12k - 4 = 0\)

Solving the quadratic equation \((7k - 2)(k + 2) = 0\), we find:

\(k = \frac{2}{7}\) only (or 0.286)