The Hardest Checkmate in 2 moves Puzzle

It is White to Play and checkmate the black king in only 2 moves. I will be discovering each move one by one. Qc6+ is met with Kf8 then Qxa8+ then Kg7. Qa4+ forces the king to move to f8 and no checkmate on that move so is the same with Qb5+. Qc8+ is obviously not going to work because black can simply play Rxc8. Qa1 is the answer but black can castle queenside. Clue: We need to figure out what was black's last move that could have lost the right to castle. If the king moves to d8 or f8 then Qh8# is checkmate. 

So far as I can see, there's a mate in 2 only if black has lost the right to castle. Working out which move caused that shouldn't matter.

The only move(s) that black could have played last were either by the king or the rook, since the remaining pawns are on their original squares.  Thus black can't castle because one or the other piece must have moved.

That is correct. There are no other moves besides the king and the rook black could have made on the last move.  This was not taken from an actual game.