My theory is that Hoid is the cause of whatever split Adonalsium and the "element" he mentions is either what enabled the split, or the centerpiece of Adonalsium, which would enable it (him/her?) to rejoin.
Why would he split it just to rejoin it again?
Maybe he didn't mean to; it could have happened by accident. I don't necessarily buy this theory, but it could be confirmed by the epigraph of chapter 14: "Let me first assure you that the element is quite safe. I have found a good home for it. I protect its safety like I protect my own skin, you might say."
I think the fact that he compares it to skin might be an important indicator that the element has something to do with a body. This also might be evidence to suggest that Hoid is Adonalsium and he protects the element in a similar way to Ruin protecting the Atium (which was his body).
Please let me know if I got any of my references wrong.