Kenpachi has a shikai sword. The only way to obtain a shikai sword is by gaining knowledge of your zanpakuto's name. There's no other way around it.
The Shikai (始解, Initial Release) is the second form (or first "upgraded" form) available to a Zanpakutō. To activate it, the Shinigami needs to learn the name of their Zanpakutō. This is not as easy as simply picking a name, as the living spirit of the Zanpakutō already has its own name. Therefore, the Shinigami must be able to communicate and harmonize with their Zanpakuto
He has a zanpakuto in shikai form meaning that at one point and time he in fact did know his zanpakuto's name. Right now, he has no knowledge of his swords name. One thing that could explain this is what's called Post-traumatic amesia particularly anterograde amnesia variation. Post-traumatic amnesia is usually caused by a head injury. If you looked at Kenpachi, there's a huge scar running down the left side of his face. It's never explained where the scar is from. He even had the scar before he fought Ikkaku. That's how long he's had that scar. With the anterograde variation it's basically the inability to remember something from your past. Kinda like repressed memories but slightly different.
So possibly, way back in Zaraki's past, he was struck with a huge blow that caused him to have a form of amnesia leaving him unable to remember what his zanpakuto's name is.