I may be solving this wrong, is black just suposed to get a king, but no draw? To get a king I believe this is the solution.
9-13, 23-18, 2-7, 24-20 (if 18-14, 7-11, 14-7, 11-16 ect. to crown, or if 18-15, 12-16 crowns.) 7-11, 27-24 and this should be a clear path to th king row after pitching 11-16 and 10-15. If 27-23 is played there instead of 27-24 the same pitches will clear the path still.