Sorting using a black box
I think your question was addressed in Beigel and Gill's paper
"Sorting n objects using k-sorter" from 1990 and the abstract of the paper says it all:


  A k-sorter is a device that sorts k objects in unit time. The complexity of an algorithm that uses a k-sorter is defined as the number of applications of the k-sorter. In this measure, the complexity of sorting n objects is between $\frac{n \log n}{k \log k}$ and $\frac{4n \log n}{k \log k}$, up to first-order terms in n and k.