Hoo boy this game is tough. I beat it (and even tried to clear the you win screen, ha!) but man.
First things first, move the cat the left side of the screen. Part of what made this game so difficult was that the player has such a short amount of time to react to what's ahead of them.
Second, buffer jumps. If the player presses the jump button a few frames before landing, they should still be able to jump on the frame that they land, or else it feels like the game ate the input, which feels awful from the player's perspective.
Third, make it easier. I'd do this by slowing the player down, or making the platforms on the stairs slightly longer. The point of game development isn't to beat the players. Everyone will have more fun with a game if they can beat it (or make any amount of progress).
Those things aside, I loved the music, and the backgrounds were goofy. It's a nice game, but definitely needs some polish.