Surprisingly enough, this was extremely easy! The Demon King has 2 phases and both have very simple attacks you can avoid with little trouble.
The battle will start with the Demon King shooting two fireballs in three patterns at random: side by side, with enough space for you to stand between and finally with a lot of space between. You'll want to run in the opposite direction and jump between the gaps. As you damage the Demon King, he will become enraged and the speed at which he fires at you increases.
Avoid the fireballs until a rocket appears and make your way to it before it takes off on its own. You're going to be hanging on to the rocket avoiding the same type fireballs until your rocket flies into the Demon King and explodes. As you're falling you'll want to attack the him as much as you can. Remember every hit gives you a jump, so chain those jump attacks and stay close to him as you fall. Once you land on the ground, you'll repeat this process until the Demon King transforms into his second form.
During the second phease the Demon King, now a generic floating eye from every RPG ever, will place energy balls on both sides of you. He will then detonate one at random sending it across the screen in both directions into the others and simultaneously detonating the others. Do your best to jump or duck under the energy coming your way until it's time for you to ride the rocket in the air.
Once you're flying on the rocket, the Demon King will be below you this time. He will place several energy balls evenly across the screen and detonate them similarly to what he did when you were on the ground. Move quickly to the closest available safe spot as he detonates the energy balls. Once you've successfully evaded his attacks 4 times, the battle will end abruptly, in victory.