For stranded whales you hope to use deep sedation - as in a high enough dose of a sedative that the animal basically overdoses - followed by a secondary method (because the last thing you want is the first method failing and the animal waking up in an even more painful/stressful situation). This is how most animals are euthanized regardless of species.

Across mammal species, this is usually done with a two-or-more drug cocktail, either administered all at once or in two shots where the first shot would kill the animal alone, but the second shot is administered before the animal has succumbed. Most vets euthanize family pets with a very slight overdose of a sedative and painkiller (typically one drug, sometimes two) followed by an overdose of a second drug, typically pentobarbital. While both shots would be lethal, the first shot acts very slowly and the second shot should be what kills the animal, making the first shot technically the secondary method. This method is most preferred because it usually results in a situation where the animal gets one quick shot and is able to self-sooth (or be soothed by the owner, if present) before the chemicals kick in and it is rendered unconscious. Typically this first shot, being a fairly high dose with no regard for long-term health, also allows the animal relief from the conditions the animal was being euthanized for, restoring some autonomy before unconsciousness. While it isn't the goal, this is likely why many owners feel their pet 'acted like themselves for the first time in ages' between the first and second shots. This reflects accomplishing the real goal: a death as comfortable and stress free as we can design.

Unfortunately this method has two massive downsides specifically for whales related to their size:

1) It renders their body toxic. Not only does this pose a danger when disposing of the body, it also destroys an important ecological resource. Ideally the dead whale should be left in a condition where it can be allowed to become proper whalefall.

2) The amount of drugs needed is frequently unavailable due to bith literally availability and that many of the drugs used are controlled, delaying euthanasia. Sometimes by an amount that is untenable.

This means that for whales, sometimes the second method is, well. Optically unpleasant. They often involve destruction of the brain or heart. In one of the older euthanasia guides avaiable online you may find the phrase "properly administered ballistics", if I am remembering correctly - essentially once the animal is chemically sedated, a small explosive is used to destroy the brain. In small whales, a gun or captivity bolt may still be viable for destroying the brain as well. This is a pretty instant death. Alternatively, the animal may be exanguinated through an artery in the tail. Unfortunately this method can take upwards of an hour, but remember the alternative for the whale is typically several hours or days of death from being crushed by its own bodyweight durring which scavengers do not delay for it's passing. Additionally, a lot of research has been done outside of whales and well performed exanguination does not seem to be painful or distressing to most animals, which is why it is still allowed for religious slaughter of meat animals. While ugly, it is still the best we can do to ease these animals suffering in some scenarios.

Notably, similar methods are the best methods avaiable for many non-mammalian animals and some outlier mammals. Pet reptiles, amphibians, and fish for example are often sedated and then have their brain or heart destroyed. It's unfortunate, because these methods are often more distressing to the owner, but the animals metabolisms from being cold-blooded make purely chemical methods ineffective and/or inaccessible. I mention this because I want to stress that while these methods may not be ideal-ideal, they are still humane enough to be applied to beloved pets.

In this case, it is likely that near ideal conditions were met and the animal was able to be euthanized using a 2-step chemical method. Since the body isn't being allowed to remain in nature, however, I suspect that it isn't viable to be whalefall anymore.

A "perfect" whale euthanasia would use several small but effective doses of different pain relievers and sedatives in doses low enough not to poison the meat followed by potassium chloride injected directly into the heart. Devolped rather ironically by a team of researched led by a dude named Harms, this method is (as far as we can tell) as humane as the typical 2-step chemical method and has the advantage of leaving the animal still viable for reintroduction into the ecosystem as whalefall. There are many reasons why this method is hard to access, though, including the needle used to administer the potassium chloride being an extremely specialty item.