It's what happens in Chapter 30. The offer to leave him alive is that he'd surrender and Harrington had arranged with the relevant authorities that he wouldn't be executed for his crimes. He'd have been in prison, but alive. Unstated in the text is that this means there was potentially a way out: people escape from prisons or arrange to be released, after all, or even before their trial. When he subsequently detonated one of his bombs and nuked a small town just to prove how serious he was, Harrington realized there was no way she could let him live.
She no longer doubted it; Warnecke was insane. Not in the legal sense of being unable to recognize right or wrong, but in a far deeper, more fundamental sense. He simply didn't care about right or wrong, and his casual mass murder only reconfirmed her earlier decision. Whatever happened, he could not be permitted to escape to do this again. Because that was the real crux of it. He would do it again, or something just as terrible. Again and again . . . because he enjoyed it.
"We can't—I can't—let him go," she said. "He has to be stopped, right here, right now."