The Kraken, according to the monster manual, can "Understand Abyssal, Celestial, Infernal, And Primordial But Can't Speak, Telepathy 120 Ft." The Kraken also does not have hands. It seems as if the Kraken can barely cast any spells at all. Can these factors be somehow overcome or explained away, ideally whilst staying within the basic 5e rules?
If you can’t speak, you can’t perform verbal components.
The rules for verbal components say:
Most spells require the chanting of mystic words. The words themselves aren't the source of the spell's power; rather, the particular combination of sounds, with specific pitch and resonance, sets the threads of magic in motion. Thus, a character who is gagged or in an area of silence, such as one created by the silence spell, can't cast a spell with a verbal component.
No speaking, no mystical words.
The DM can do whatever they want, so make it psionic.
The DM can make monster’s do whatever they want to, and don’t even have to justify it with logic, especially for something like a boss monster. Just give it innate Spellcasting:
A monster with the innate ability to cast spells has the Innate Spellcasting special trait. Unless noted otherwise, an innate spell of 1st level or higher is always cast at its lowest possible level and can’t be cast at a higher level. If a monster has a cantrip where its level matters and no level is given, use the monster’s challenge rating.
Monsters with innate Spellcasting often have the ability to cast their spells without components because they’re inherently magical creatures. Just make you kraken an innate spellcaster (often called psionic).