Their relationship can vary wildly depending on writer or adaptation, but the two most common explanations seem to be either Lex respects the Joker for his unpredictable nature and unique genius, or, as you said, he cannot omit him. Did some digging on the topic of their relationship, I think what you are thinking of is the Joker killing the entire LOD in Infinite Crisis. Deaths Lex set up Joker to do since his son took power from him.
Then in another story, Lex will randomly talk about how he and Joker would be unstoppable together if they combined their smarts. That Joker is a brilliant mind from a certain view. Most adaptations seem to go with Joker being useful, particularly the DCAU, which I prefer as it avoids Joker-wank, as Lex should theoretically be able to just get rid of him with decent success. Not like he has powers to really put up a fight.
https://youtube.com/watch?v=TEtb4pcae6wOn more rare occasions, they are pretty much made to be outright friends. The LEGO adaptations in particular take this view, along with Justice League Action, kind of. Even some comic excerpts play into it.
Personal view, Joker really is Lex's only "friend." The man definitely respects the clown and values him. I think Joker hits the sweet spot of being someone Lex can work with given that Joker doesn't exactly look to completely take over his shit, at least not initially like a Grodd would. I also think Joker likes Lex, arguably has been shown to be even more affectionate towards him. The two certainly piss each other off, but they really are the only ones who can really understand the worldview of the other. As I put in the original post, they are both nihilists, but with completely different takeaways: