How many minimal $k$-cycles can you put in a graph?
If $k$ is divisible by 2 we can take a clique on $n/2$ vertices and subdivide each edge once.
This graph will have ${n/2 \choose k/2} \cdot (k/2-1)!/2$ cycles of length $k$.

Hence the total number of cycles is $\frac{1}{2} \sum_{i=k/2}^{i=n/2} {n/2 \choose i} \cdot (i-1)!$