That is tough. I tend to agree with the members who suspect that she'll never, ever give up anything to investigators. But I like what you say about investigators playing the suspects by appealing to whatever their "needs" are. I'd suggest that LS's greatest need in this situation (in addition to her need for control as you say) is to appear to be a victim, to have been unfairly treated. So maybe a detective could suggest that what happened to Gannon could have been an accident, and that (since LS is always treated unfairly and blamed) she was afraid that people (AS, LH, big bad police) would blame her for it. Even though she was completely blameless and even heroic in her efforts to save Gannon. Therefore, she hid Gannon's body.
She might not fall for it, but she's about half as smart as she thinks she is (at most). I could see her ego being so needy that she'd fall for it. If not from a detective, possibly a fellow inmate? IDK. Just some ideas.