The cocoon itself doesn't have health or take damage, this boss fight involves destroying the 6 vines that surround the cocoon to drop it off of the ceiling. There are 3 vines on each side, and each vine takes 3 missiles to kill, meaning you will need to fire 9 missiles on each side, for a total of 18 missiles required to finish the fight.
This fight has two different strategies that depend on whether or not you have Hi-Jump Boots when you enter the fight. In 100% you will always have Hi-Jump Boots, and in Any% it is optional if you prefer to pick up the Hi-Jump Boots before or after the fight. In Low% runs you don't have enough missiles to do the non-HJB strategy so it is slightly different and that fight ultimately depends heavily on drop luck.
The first vines
No matter what category you're playing, the first vine on both sides of the cocoon is reachable by firing 3 missiles straight up from the platform. The places where you can stand to hit the vines are pretty lenient but still precise enough that you may have some trouble getting into the right spot until you get a feel for it with some practice.
With Hi-Jump Boots
There is a very small spot on both sides of the platform below the cocoon where you can jump straight up and fire two missiles into the vines. A optimal cocoon fight with HJB involves getting on the right side quickly and destroying the first vine, and then jumping straight up 3 times to fire the last 6 missiles into the right side. You can freeze the ripper during this process to keep it out of the way. Once the right side is cleared you can run to the left side and do the same thing.
Without Hi-Jump Boots
The fast strat without HJB is a little more difficult because there's a lot more inputs going on. You want to enter the room and freeze the ripper as soon as you see it and immediately fire the first three missiles to kill the first vine on the left side. Next you want to wall jump up the left side high enough that you can fire two missiles straight into the left vines and a third missile angled up as you're falling. Do this twice and you've cleared the left side at the same time the ripper unfroze and moves into position on the right side. If done correctly, you can refreeze the ripper on the right side in a position where you can stand on it and fire all 9 missiles into the right side.
Because you only have 5 missiles in 9% runs there isn't as much emphasis on how quickly you can get your shots in, and the speed of the fight is more dictated by how lucky you are with drops. A good start is to go ahead and kill the right vine while you wait for the ripper, and freeze it on the right side. Farm the drops and kill the vines however you want to, the only way to speed up or slow down the fight is by being efficient with your ammo and not wasting drops or missing shots.