We can consider both $ n $ and $ k $ as multiples of their greatest common divisor: \begin{align*} n &= n'\cdot\gcd(n,k), \\ k &= k'\cdot\gcd(n,k), \end{align*}where $ n' $ and $ k' $ are relatively prime integers. Then $ \mathop{\text{lcm}}[n,k] = \frac{n\cdot k}{\gcd(n,k)} = n'\cdot k'\cdot\gcd(n,k) $, so $$\frac{\mathop{\text{lcm}}[n,k]}{\gcd(n,k)} = n'k'.$$We have $ \frac{n'}{k'} = \frac nk $. So, we wish to minimize $ n'k' $ under the constraint that $ 5<\frac{n'}{k'}<6 $. That is, we wish to find the smallest possible product of the numerator and denominator of a fraction whose value is between 5 and 6. Clearly the denominator $ k' $ is at least $ 2 $, and the numerator $ n' $ is at least $ 5(2)+1=11 $, so the smallest possible value for $ n'k' $ is $ (11)(2)=\boxed{22} $. Note that this result, $ \frac{\mathop{\text{lcm}}[n,k]}{\gcd(n,k)}=22 $, can be achieved by the example $ n=11,k=2 $.