Kth smallest in a row-sorted matrix

You are given an n x n matrix whose every row is sorted in ascending order (and, in the classic version, every column too). Return the k-th smallest element in the whole matrix, counting duplicates.

matrix = [[ 1,  5,  9],
          [10, 11, 13],
          [12, 13, 15]]
k = 8   ->   13

Aim for $O(k \log n)$ time and $O(n)$ memory.