Oh, yes, I gave a fairly complete answer at:

A hierarchy of k-highly composite numbers

There is a procedure iniated by Ramanujan that gives a sequence of particularly large values of $$\frac{\sigma_k(n)}{n^k},$$ essentially by taking any $\delta > 0$ and finding the $n = n_\delta$ giving the maximum of $$\frac{\sigma_k(n)}{n^{k + \delta}},$$ and choosing the largest if more than one $n$ gives the maximum. The recipe gives a recipe for the prime factorization of $n = n_\delta.$ Once one has a value, it follows that the original $\frac{\sigma_k(n)}{n^k}$ is larger than for any $m < n.$

There is a fair amount of work involved in interpolating these values into a bound of the type you quote. So, as far as I know, it has only been carried out in entirety for $k=0,1,$ where the former refers to the raw count of divisors.

EDIT: I found the complete answer for $k=0,$

What is the lower bound for highly composite numbers?