Okay. This may not look like the greatest Emerl skill combo ever, but trust me I'm an expert. I beat this entire game in four days. Here is the Emerl skill combo you should use to fight Eggman on the Death Egg:

Jump Skill: Rouge Flight
Air Action: Amy Double Jump
Heal Skill: Recovery Mode
Air Attack: Agent Eagle

The others don't really matter. DO NOT equip a Combo Card, Eggman hardly ever lands long enough for you to use one. When the battle starts, double jump over Eggman. Because you have the Rouge flight, you will be higher, and you will descend about the same speed as Eggman. When you are directly above him, keep using the Agent Eagle. It doesn't take too many hits to take him down. After he is KO'ed wait until you are on the ground and use Recovery Mode to regain your Health back really fast. Then repeat until he is done.