If kill is really in quotation marks, then I don’t think Azami means it in a literal sense, which means there‘s only one other thing I can imagine she means and that’s why she needs Hinako.
Sure, she hurt Hiori before, but she didn‘t die and she eventually healed. That’s not enough to satisfy her vengeance. She wants Hiori to experience the same kind of pain she did, to want to die but be unable to. Before, that was impossible, but now, there’s someone that Shiori actually cares about, someone she wants to live. What better way to “kill” someone than to go straight for their “heart”.