I've been spamming the shit out of Kayn, and have already played him to mastery 5. The purpose of this post is so that hopefully someone can see this and relay the feedback to the balance team.
The champion is fun and the concept is pretty sick, with his thematic and lore really matching his quotes and in-game personality, so you've hit that nail on the head. His kit does generally feel satisfying to play as and against, with apparent counterplay for both forms. The key issues he has at the moment are as follows:
1) His pre-transformation.
Currently his pre-transformation form is fairly weak, and for good reason as this is meant to be before his form switch. The problem with this is that he needs to fight to get his formswitch. He is like kindred, very weak early however he needs to fight whilst he is weak to ensure safety in the midgame. I think buffing his early game stats will make him less feast or famine.
2) His shadow assassin form.
This form is a bit weak. It's damage is good, and i think it's weakness is partially derivative of assassins being weak in general at the moment, however the form is almost never a better choice than Darkin, even in games with lots of squishies. The ultimate is the culprit of this. It offers pretty much nothing over the Darkin's counterpart. The Darkin gets a nice big heal so that he can stay in the fight, whilst also doing a hefty chunk thanks to the max health damage. this being paired with the Darkin's extra natural survivability makes it so that popping out right next to your victim isnt so bad; you really want to be in the thick of it soaking up damage and reaching their backline. In shadow assassin form, this isn't exactly what you want. You can only really use this ability in duels, or as a last resort before death, as your victim can just walk back into their team, get shielded and healed up, and explode you as soon as you come out. Since the damage isn't modified it doesn't do much at all even to squishies, and the passive refresh isn't useful when you just get cc locked as soon as you exit. I have a few suggestions for this ability:
-Increase the damage in shadow form, or have it as missing HP damage instead so it can be used as an execute.
-Make it so that the ultimate leaves behind a shadow which you can hop back to after you've left the ult form, similar to Zed's.
-Make it so that using your ultimate has increased exit range, and if you collide into a wall upon exiting your victim, you immediately get put into your E.
This makes it so that you don't just always see darkin forms, as that isn't what the champion is about.
Other than that, im loving him so far. Hopefully these suggestions are useful/get passed on.