illegalest move on Earth
White is in checkmate. He has no legal moves by art 1.4.1
... . Leaving one’s own king under attack, exposing one’s own king to attack and also ’capturing’ the opponent’s king is not allowed .
Your solution is not valid. The game has terminated under basic rules. (And 1.Qe4#, which is just as legal, would seem to be preferable. After 1.Kxe5 White can no longer checkmate.)
In fact "White to move" is also incorrect by art 1.3
A player is said to ‘have the move’ when his opponent’s move has been ‘made’.
Under art 4.7 a move is 'made' only if it is legal and no possible last Black move would be legal.