jmc

We can consider both $n$ and $k$ as multiples of their greatest common divisor: $$\begin{array}{rl}n& =n^{\prime }\cdot \gcd (n,k),\\ k& =k^{\prime }\cdot \gcd (n,k),\end{array}$$where $n^{\prime }$ and $k^{\prime }$ are relatively prime integers. Then $\mathrm{lcm}[n,k]=\frac{n\cdot k}{\gcd (n,k)}=n^{\prime }\cdot k^{\prime }\cdot \gcd (n,k)$, so $$\frac{\mathrm{lcm}[n,k]}{\gcd (n,k)}=n^{\prime }{k}^{\prime }.$$We have $\frac{n^{\prime }}{k^{\prime }}=\frac{n}{k}$. So, we wish to minimize $n^{\prime }{k}^{\prime }$ under the constraint that $5<\frac{n^{\prime }}{k^{\prime }}<6$. 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 $k^{\prime }$ is at least $2$, and the numerator $n^{\prime }$ is at least $5(2)+1=11$, so the smallest possible value for $n^{\prime }{k}^{\prime }$ is $(11)(2)=\boxed{22}$.

Note that this result, $\frac{\mathrm{lcm}[n,k]}{\gcd (n,k)}=22$, can be achieved by the example $n=11,k=2$.