Constructing a Divide and Conquer Algorithm
I believe you have misunderstood the question. The magical merge algorithm merges two lists of size $m$ in time $m^{1-\epsilon}$ for every possible $m$ (of course, such an algorithm is impossible). Since merging usually takes linear time $O(m)$, it is always better to replace it with the magical merge algorithm. You then get a different recurrence for your magical merge sort, and the running time should come out linear rather than $O(n\log n)$.