EVIL:
For me, Kekkai Sensen is my favorite anime this Season. Don't care if others don't find it good or entertaining (as if anime title that can please EVERYONE exists in this world, anyways).
After watching eps 6, I have some guesses regarding White and her brother (Black, aka "King of Despair"). Given that they're anime-original, I doubt you can think this as spoiler, though...
In eps 2, she said that she's a ghost, yet other people than Leo can also see her and interact with her (end of eps 5, she talked to one of the doctor). I don't think she's lying about it, though. I believe she's a ghost that possessing someone's body (in this case, the original 'White' girl). This based on how she reacted on Leo's photo about certain graveyard in eps 2-3. Given that she's confined in hospital since 3 years ago, I assume original White has some serious illness that make her doctor forbids her to leave the hospital (whether her illness has been cured or not when "ghost-White" possessing her, can't say).
He appeared first at the end of eps 1, then met both Leo and White in eps 4, and met up with Fehmt in eps 5. In eps 6, he appears to have good relationship with White, in contrast with her reaction to him at eps 4. Based on that and how he said that he never met Leo in subway before (eps 4), I think he's the same as "ghost-White"; a being that possessing human's body for some reason. His real body is that red-eyed blondie in blue coat that appears in OP&ED song. I think he's kinda different than "ghost-White" since he's one of the Kings, though (this also emphasizes of why White cried when watching movie with Leo at eps 5. Now, unlike White, character/personality of original Black seems intact and appears when King of Despair is not taking control of the body, hence why he doesn't remember about Leo and also explain why White is not acting cold toward him in eps 6.
If my guesses above are right, then my other guess about White-Black's fate at the end of this anime might be also hit the mark (not gonna say it now, though). But if I'm wrong, then those are merely wild guesses from me, hah.