Barry's move to STAR Labs was accompanied by an entourage of visitors, much to the apparent consternation of Harrison Wells. He clearly wanted to work in private, but he couldn't very well hide Barry away from his family and friends, and so he was forced to accommodate the constant parade of people coming in and out of the Cortex to see him. As a result, those people rarely if ever actually saw whatever work Wells and his team were doing to fix Barry's condition, and so had to take it on faith that it was happening.
Barry's constantly rotating parade of visitors included Kara, Henry and Nora, Joe, and Iris, as well as the occasional colleague from the CCPD. As the weeks passed, Henry was eventually forced to return to work, which he did only with the knowledge that there were plenty of other people watching over Barry, who would let him know if there was any change in his condition, either positive or negative.
This pattern continued for the next several weeks. Eventually, Kara started coming in to the office every day once again, something that she, like Henry, was only able to do because she knew there were others- namely Joe, Iris, and Nora- watching over Barry. All the same, every spare moment she had, she spent with him at STAR Labs.
Upon one such instance, she walked into the Cortex to find a woman she didn't know sitting alone at Barry's bedside, Iris and the others being elsewhere, at least for the moment. She wore clothing of the type that toed the line between formal and casual and glasses with two-toned rectangular frames, her shoulder length blonde hair pulled back in a neat, no-nonsense ponytail. She must have just come from work, as she had an ID badge around her neck that, in the brief glimpse Kara caught of it, identified her as an employee of Queen Consolidated.
"You must be the girl Barry met in Starling City," she surmised aloud. "The one who works with the vigilante."
"Hmm?" the woman asked, turning her head toward Kara, her earrings swinging in time with the motion. "Oh. Yeah. I'm Felicity. Felicity Smoak. And you are?"
"I'm Kara," Kara answered. In response to Felicity's questioning look, she clarified, "Barry's sister." Felicity nodded.
"Barry never mentioned that he had a sister," she muttered.
"It must have slipped his mind," Kara replied, sitting down in one of the unoccupied chairs at Barry's bedside. "You know, what with everything else that was going on while he was in Starling." Felicity nodded again, and they both sat in silence for a few minutes before she asked "Barry didn't… tell you who the vigilante is, did he?" Kara shook her head.
"No," she said. "He didn't. I asked, but he told me that it's not his secret to tell." Felicity looked relieved.
"Good," she said quietly. "That's good." They sat in silence again, until the sound of a ringing phone disturbed the quiet.
"What's going on Oliver?" Felicity asked, answering it. "What do you need?" Kara wasn't trying to listen in, but with her super sensitive hearing, she couldn't help but overhear the person on the other end- Oliver, apparently- reply "Why do you assume I need something? I can't just call you because I want to talk to you?"
"You could," Felicity said. "But you don't."
"Well, this time I am," Oliver replied. "I wanted to see how you're doing. Are you holding up alright?" At that point, Kara deliberately blocked out Oliver's half of the conversation, his words making her realize that this was a personal matter that shouldn't be listened in on by a stranger.
"I'm fine," Felicity said, an emotion in her voice that Kara couldn't name, not that she was really trying to in the first place. "Are you sure you guys don't need anything?" Whatever Oliver's response was, it must have brought their conversation to a close, as Felicity hung up her phone a moment later. Then the two women sat in silence once more, until they parted ways at the end of the day.
Kara saw Felicity only sporadically over the next few days, as she tended to trade off with Nora and the Wests and thus was only present when they were not, and when she did see her, she noticed that she seemed distracted, pulled in two different directions at once, her focus split, alternating between sitting quietly at Barry's bedside and having hurried, terse phone conversations with her boss, her half of which seemed to have to do with something other than the normal day to day of her job. Eventually, Felicity's work called her back to Starling City, and Kara thought no more about what exactly those phone conversations might be about.
As months passed, Kara began to accept this pattern of going to work and then visiting her brother before going home to sleep and then waking up to do it all over again the next day as her new normal. She became well acquainted with the members of Wells' team- though not Wells himself, as he was an extremely private, closed off type of individual- and saw more of Nora and the Wests than she had in quite some time. Things continued on that way, mundane and routine, until Kara stumbled into something that threatened to shake up what they all now thought of as "normal".
She was walking toward the Cortex when she overheard Iris talking to Barry. That in and of itself was nothing unusual- she'd heard that coma patients were aware of everything that was going on around them, and so she often talked to Barry when she sat with him. It was what she was saying that caught Kara off guard.
"Detective Pretty Boy asked me out on a date today, and I actually said yes," she heard Iris say as she neared the entrance of the Cortex. "See, that's the kind of bad decision I make when you're not around." Her voice quavered, like she was about to start crying, and Kara stopped in her tracks. After debating with herself for a few minutes, she turned and left, deciding to give Iris her private moment with Barry, as she clearly hadn't meant for anyone else but him to hear what she'd just said. Even so, Kara found herself thinking about it the whole way home. Iris may have just meant that she'd made a poor decision without Barry there acting as a voice of reason- which, among the three of them, he often was, at least when it came to relationships- but Kara couldn't help but wonder if maybe, just maybe, Iris' words meant that Barry's crush on her may not have been as one sided as he and his sister both thought.