According to the Wikia, Yachiru disappeared leaving her belongings after the fight with Gremmy. She is said to be "gone" by Kenpachi in the penultimate chapter of the original series.
I use "original" because there might be extra chapters announced as the other big announcement.
So, what happened to her? Was she part of his Zanpakutou or something?
The very last page of Bleach #668 gives a few clues.
Yachiru suddenly appears out of nowhere and says to Kenpachi:
Silly goose, if you would just use me properly, there wouldn't be anyone you wouldn't be able to cut down.
She then touches him, which causes:
him to get a power boost, which he wonders about.
She ends the exchange with explaining to him:
That power is what everyone's been calling "Bankai".
All of these suggest the following:
Yachiru is Kenpachi's Bankai. He is unable to call out his Bankai anymore (he was never able to, consciously - the original appearance may have happened subconsciously), therefore, Yachiru is gone.
Alternatively:
Yachiru is his Zanpaktou spirit, again, one he has never been able to consciously communicate with. Though, I can't explain her disappearance if that were the case.
It would most likely because that she is a Zanpakuto who had split herself and physically manifested as a spirit. Kenpachi and Coyote Starrk are similar as they have their souls/powers split resulting in Yachiru and Lilynette.
Like many people who already mentioned it, when Yachiru disappeared, she most likely fused back into Kenpachi's Zanpakuto just like Starrk did with Lilynette.
Short answer: Yachiru is with Kenpachi; as his Zanpakutou's personality.
Long answer:
I cannot cite manga pages and chapters but if you follow the story, you should remember these.
Yachiru has always been beside or around Kenpachi as a human. Kenpachi's Zanpakutou has never had a name. There was once when Kenpachi lay on the ground trying to ask his Zanpakutou what its name was; no response. Kenpachi's mode of fighting has been brutal which damages his sword.
Fast-forward to Yhwach's arc as Soul King, Yachiru makes Kenpachi know that the power he feels is what others call Bankai. This is after she fused with his Zanpakutou.
I believe Kenpachi would now be able to call out his Zanpakutou and her name would Yachiru.
With the release of The Thousand Year Blood War, felt like rereading the manga. In chapter 577, just before vanishing, she is recalling her first meeting with Kenpachi, personally, I think she might be caught in a time loop.
Meets Kenpachi, spends years with him till that final battle, thinks strongly about the first meeting, and since some residue of Gremmy's power was affecting her, returns to that original meeting as that Yachiru and repeats.
I think Yachiru is a physical manifestation of Zaraki's power. When Gremmy said that "he would imagine a world without imagination", Yachiru also vanished.
Yachiru was Zaraki's true spirit. As for Nozarashi, it was the Zanpakuto of a dead Shinigami that Zaraki took the sword from. It's explained by Ōetsu Nimaiya.
Remember that Zanpakuto's real name is Asauchi. If he dies there, Zanpakuto and the spirit inside still reside there. So, Zaraki had two spirits.
It isn't a matter of debate. Yachiru is his Zanpakuto spirit. Her talking about herself as his Zanpakuto is fairly evident. The manga shouldn't have to have her say, "I'm your Zanpakuto, Kenny!"
Yachiru is not Kenpachi's Zanpakuto or the spirit of his Zanpakuto. She is a Shinigami who has her own Zanpakuto as shown in her battle against Gremmy 'the Quincy who could make things he imagined real'. She is a soul from Kasajishi district of Rukongai. She may or may not have been near Kenpachi before he gained his Bankai as she disappeared immediately after seeing Kenpachi slice the meteorite.
The only reasons I have for Kenpachi seeing Yachiru before releasing his Bankai are:
