Kalle Demos can be killed two different ways, the intended way using the boomerang and your sword or with bombs.
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 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.
If you have a bottle, then you can also one shot Kalle Demos using Forest Water. This is slower than defeating him with the sword due to slow animations, but is very useful if you come to Forbidden Woods without a sword.
If you use all of your bombs during this fight, Kalle Demos can still be killed with the use of the infinite bombs glitch.
Despite Kalle Demos being protected by the massive flower encasing it, a bomb's explosion radius is actually large enough to hit his head while it is in the flower provided you throw bombs from under him properly. The timing and positioning for these throws are kind of tricky and take time to get used to. Since you are able to kill Kalle Demos purely with bombs, this allows you to skip the boomerang as the only thing the game requires you to get the boomerang for is this boss fight. Generally a fight with bombs is faster than a fight with the boomerang, so even in a 100% run where you will have the boomerang, it is best to use bombs.
To make a bomb hit him through the bottom of the flower, stand roughly under the bottom flower and back up a little bit. You want to throw the bomb shortly before it explodes. If timed right you will hear Kalle Demos take a hit. Each bomb deals four damage so you must hit him with four bombs to deal 16 damage and finish him off.
For some strange reason Kalle Demos does not have a hit box for a very brief moment when the fight starts. Not only can you time bomb throws to hit the head, but you can throw a bomb up in to the flower while there is no hit box. The bomb stays in the flower next to the head, thus allowing for an easy hit, and letting you start the timer on your second bomb by pulling it almost immediately. The easiest way of achieving the early bomb throw is to hold L + up before/right as the fight starts, and time the bomb pull/throw. This can be seen in the video below.
It is also possible to get bombs stuck in the tentacles. This has a similar result to getting the first bomb stuck in the flower next to the head. You are able to get a relatively easy hit and can pull your next bomb almost immediately to start its timer. The positioning is a bit difficult to get used to. This can be seen in the video below with the second bomb.
An unfortunate side effect of killing Kalle Demos with bombs it that is possible to pick up the heart container during the cutscene afterwards. This is, for the time being, completely uncontrollable. It is not a common thing so you shouldn't be too worried about it happening. This is a problem for low% runs as in order to skip the boomerang you NEED to use bombs on this boss, and that ultimately means the slight chance of picking up the heart, leading the the end of your run.
Due to the wacky nature of Kalle Demos' death scene after killing him with bombs, Link can get stuck on the boss remains when he tries to walk over towards Makar. This unfortunately causes an infinite loop of the game constantly trying to progress the cutscene when it physically can't due to Link being stuck. This is commonly known as a softlock, meaning that you can pretty much do nothing to recover from it other than turn off your console. The chances of this happening are extremely slim.
For what ever reason, this can also happen after killing Kalle Demos normally. What causes it is still unclear. Again, this is a very rare occurrence.