I'm two issues behind on Harley Quinn currently, so this may already be dated, but what the hey.
The patient who left Harley her current digs is also the person who put the hit out on her.
I don't know Harley's full backstory in the new 52, but we know she for certain she worked at Arkham. She was primarily interested in the Joker, but I'd be really surprised if he was the only inmate she saw. Can't she even as shoddy an operation as Arkham letting some newbie start in on their most dangerous patient right off the bat. Arkham is full of dangerous people, though. Ones Harley may have crossed while working for/pursing Mistah J, or as part of the Suicide Squad. Or they may simply not like her. We're dealing with people who don't have the firmest grip on sanity, so there's no guarantee their reasons are anything I can fathom.
But it does seem remarkably convenient that Harley's storage unit blows up, and immediately, here's some legal eagle telling her she's inherited property. And once she gets there, people immediately start trying to kill her. It would be a pretty sound plan. Take away her living space (and maybe kill her). If she survives, now she needs a place to stay, and you provide one for her. It also means there's a solid lock on her location, which makes it easier to send other killers, or just to keep tabs on her. And, since she was told the property was "left" to her, she'll never think to look into who the patient was, because the patient is dead. They can't have put a bounty on her head.
I guess there's a chance it could be the Joker behind it, though that would involve him putting more thought towards Harley than usual. Maybe Waller's trying to tie up a loose end? Again, seems a little out of character for her, but I'm thinking of the pre-reboot versions, so who know how applicable that it.