We already know that the inner loop runs in O(klog(n)). Since we've just added a loop around that, it's easy to see that the whole thing runs in O(mklog(n)) time.

This is not necessarily true for all cases. (It is for yours.) It's possible for interaction to exist between a loop and its contents, such that the total time is not m times the running time of the interior of the loop.

  for (int i = 0; i < data.length(); i++)
      bubble_sort(data);

That's a trivial case but sufficient to prove the point. Bubble sort runs in O(n^2), and it looks like we're doing that n times, so the whole thing runs in O(n^3). But that's not true, since bubble sort runs in O(n) when the array is already sorted, so the whole thing is still O(n^2). In other words, the first iteration of the loop induces a side effect that affects all later iterations.