Hello! Thank you so much for your comment! The solution to your issue is that it's possible to swing the sword downwards while holding the down button on the d-pad while in the air, and then pressing the sword button before you hit the ground. When you swing the sword downwards, and hit a spike, enemy, or even a door, you will get a height boost equivalent to a jump, allowing you to make the jump that you previously described. Similarly, you can hold up on the d-pad and press the sword button to perform an attack above you. If you're curious about any of the other abilities, there's a quick guide to using them in the description of the game. Hopefully you'll be able to surpass this obstacle now :)
-R