Kalle Demos has two phases, but both of which are very fast if done right. For the first phase you need to cut all the tentacles attached to the ceiling using the boomerang. This can be done with one throw however it is highly unlikely as it is very precise. It has been deemed TAS only but a few people have managed it. Generally you want to aim for 2-3 throws during a speedrun.
The best way to begin the battle is by backing away to give yourself a better view of Kalle Demos. To cut down as many tentacles as possible in a single throw, target one tentacle on either side of the boss, move the target reticle down and under Kalle Demos to the other side, then target as many tentacles as possible on that side. This will, if you are lucky, cut all the tentacles behind him and only leave a few in the front. These can easily be cut with a second throw. Once all the tentacles are cut from the ceiling, Kalle Demos will fall to the ground and open up, allowing you to attack its head.
The tentacles attached to the ceiling can also be cut down by using the grappling hook. To do so, you need to stand far enough away from Kalle Demos to not have the grappling hook hit him before the vines, but not too far or else the grappling hook will miss completely. Aim at a vine, throw the grapple to cut it down, rinse and repeat to cut all the vines one by one. Since the vines start recovering after a while, cutting all the vines down in a quick succession without wasting any time is needed, making this strategy really difficult to execute properly. Going in a circle around Kalle Demos as you are doing this is recommended. This strategy is required in order to skip the boomerang in an All Dungeons run.
The head requires you to deal 16 damage to it for the boss to die. For a table of weapons and how much damage they deal, go here. The Hero's Sword as you will see only deals one hit of damage. Fortunately spin attacks deal two damage instead of one, and if the enemy you are attacking does not recoil far away when you hit them with a spin attack, it hits twice. This is the case with Kalle Demos. Each spin attack will deal two damage and hit twice, totalling four damage. It is possible to do four quickspins before the boss closes back up, allowing you to deal the entire 16 damage and finish it in only one cycle.
Another option for the second phase is to simply pour forest water into Kalle Demos which doesn't change his health but simply activates his death cutscene immediately. This option is good for beginners who may not be consistent at killing Kalle Demos with the sword in one cycle, as it is only a few seconds slower than using the sword.