You checkmate a lone king with king and queen by walking the enemy king to the edge, then delivering mate with the queen supported by your own king. Use the queen to shrink the enemy king's space, staying a knight's-move away to avoid stalemate, and bring your king up to guard the queen. Once the enemy king is on the edge, place your king in front of it and give check with the queen on the last rank. The main danger is stalemate: never take away every legal move unless it is actually checkmate. Keep the king boxed but not frozen until then.