If I understand the question right, then $k$ is simply the number of divisors $d$ of $\text{lcm}(a,b)$ such that $a\le d\le b$ and $d\mid\gcd(a,b)$. (So in finding a longest chain, we may assume that $a$ and $b$ are relatively prime.)

In your example, a longest chain would be
\begin{equation}
7200\to 7560\to 7776\to 9072\to 9720\to 10080\to 10800\to 12600\to 12960\to 13608
\end{equation}
I doubt that the length can be determined without some sort of actually computing the divisors. One can interpret these divisors (via their exponents) as lattice points in a polytope, so if $a$ and $b$ have lots of divisors, one probably better counts these lattice points.