It's white.
OK. The first question is: What was White's last move? It had to have either been a pawn capture or a King move, since if the a1 Knight is white it had no square to move from anyways. The thing is, if it was the pawn capture the Black King had no way to get to g1. This means it must have been a King move from d1, as from any other square it would be a triple check or impossible double check. So we know White's last move was Kd1-d2 possibly with a capture. Now, we know black's last move was Qf1+, and it wasn't a promotion (all eight Black pawns are on the board, and the pawn would have had no square to come from anyways). So where did the Queen come from? Had it been at e2, White's last move would have put himself in check, so it was at d3 and there was a piece on d2 blocking the check that was captured with Kxd2. The piece must have been a Bishop or Knight, since the Queen and eight pawns are accounted for and a Rook would have been check. However, the dark-squared Bishop has been stuck at f8 the whole game, and can't have gotten to d2! The piece is a Knight, and since one black Knight is on the board, one is captured, and eight pawns are still on the board the piece on a1 can't be a black Knight. It's a white Knight.
The Knight on a1 might be an impostor, but the King is unsure! Knowing that it is Black's turn to move, prove what color the a1 Knight is!