Billy was certain that Frank never saw him in the stairwell, mostly due to the lack of even one attempt on his life. Madani never caught up to him either, but given her suspicions over his involvement in her partner's death, he wouldn't be surprised if she tried to (rightfully) pin this on him, too.
Failure to eliminate Frank aside, he'd still gotten one good thing out of Lewis' rampage: Karen. Karen Page, the journalist with a concealed carry permit. And, apparently, Frank's loyalty. He'd figured there was someone that Frank cared about; it was all over his face after Billy tried to put him on a boat and mentioned he didn't have any family to tie him to the city. He just hadn't known who it was. But then Frank ran into a building full of security and cops to take out a kid because he was going after Karen. He'd literally taken bullets for her, almost got blown up, and suffered some pretty serious (at least for anyone else) injuries. Yes, Frank definitely felt something for Karen Page, for all she would deny they were anything but friends.
So, after he'd finished with the NYPD and called the families of his fallen employees, he found her contact information from ANVIL's background check and pulled her gun from their safe, marking the log as it having been moved to their facility for safekeeping. He didn't know if she'd been flustered after her ordeal (which he thought unlikely) or was just too tired to care just then (more likely), but she didn't call him that night to retrieve her firearm. He gave her until the following afternoon and then called her himself. She groggily (he had woken her up) affirmed that she wanted her .380 back and asked if she could stop by the next day. Instead, Billy offered to bring it to her, since he "just so happened" to have a meeting later just a couple of blocks away. When he stopped by her apartment to return her gun, she answered the door in pajama pants and quite possibly the rattiest sweatshirt he'd ever seen. She took her gun, checked the magazine, and then tiredly invited him in for a drink.
Thus began Billy's plan of keeping tabs on Frank through Karen Page.
