I believe in Kai and Sensei Garmadon solidarity.
Like, they had their issues when Garmadon was evil, he did kidnap Nya and Kai thought his true potential was going to be defeating the man after all, but now that Garmadon isn't trying to stop them every time they turn around, they get along.
Especially when it comes to protecting Lloyd.
Like, season 3, when they find out the Overlord's plan to take Lloyd's powers, Kai (and the others) immediately freak out and are like "right, we gotta get this kid out of here" and where do they go? Garmadon's monestary.
I can totally imagine that in between when they get there and before The Art of the Silent Fist™ scene, Kai is kinda freaked out about the Overlord getting to Lloyd, and Lloyd's just like,
"Bold of you to assume that between you and my father, he's even gonna get close to me," cause, you know, Lloyd was a little cocky in season 3.
Than, after Lloyd and Garmadon go into hiding and the whole thing where he gets taken by Pythor and the Overlord happens, in between the ninja getting out of the digiverse and them making contact with Lloyd, this happens,
"But what about my son?"
Kai, freaks out and turns to Garmadon, "what happened to Lloyd? Where is he?"
Garmadon explains what happened and how he feels as if he failed, and he and Kai actually hug in a way that's like "we both need support cause the one person who drew us together is missing"
Than Pixel makes contact with Lloyd and after they pick him up from the remote village he got to, Lloyd's exhausted for obvious reasons and Kai immediately hugs him and says, "don't ever assume you're not going to get kidnapped again."
And on the way back to New Ninjago City, Lloyd ends up falling asleep sandwiched between Garmadon and Kai because there is no way those two are letting him out of their sight ever again.
In season 4, sometime on Chen's Island, they had this long conversation about Lloyd and how much they care about Lloyd, and Garmadon saying something to the effect of "if something happens where I'm no longer here, please look after my son." And Kai agreeing, thinking it's just old man talk™, not to mention, he already would fight for Lloyd to the death.
And than after the season 4 finale, Kai takes it upon himself to look after Lloyd and make sure he's okay, cause he literally just told his dad that he would do that.
So in season 5, when Lloyd gets possessed by Morro, Kai feels like a failure, both to Lloyd and to Garmadon, and he's nearly ruthless in trying to get him back.
I don't know how to incorporate the end of season 5 after Lloyd sees Garmadon in the Cursed Realm or anything involving Day of the Departed, but as for season 8...
When Lloyd gets hurt by Evil!Garmadon in Kryptarium prison, Kai holds his hand the entire trip on the way to Mistakyé, and he can feel his hatred for Garmadon come back.
Because the Garmadon he knew, the Garmadon he told that he would take care of his son, who is currently fading in front of his eyes, wouldn't never hurt Lloyd. After all, the season 2 battle was Lloyd vs. Overlord, not Lloyd vs. Garmadon.
And as Kai squeezes Lloyd's hand a little tighter, to assure him that he's there, he knows that there is no way to bring Lloyd's father back.