The opponent is losing, so he just keeps checking my king

what

´-´´

If a player can check again and again he can not be lost. It is at least a draw then and if it is a draw (aka, he can check the king but he can not mate him) it also has a name: Perpetual check.

for example in this position white is up a pawn and is threatening mate, but black to move can give perpetual check or wins the pawn, therefore the position is draw.

Losing players are going to play for a draw. It is what you do when you are losing. One of the ways to do this is forced repetition. It is up to you to make sure they can't do this and if you fail to then you have not won the game.

I have drawn games I had been losing by perpetual check and had my my frustrated opponent ask "Is that all you can do?" My reply is always that I could ask them the same question.

I would write my congressman

Check is best option in losing position.