I started biting my nails at about that age, and I desperately wanted to stop for many years before I finally managed it (mostly, anyway) as a high school senior. I internalized shame quite effortlessly, and I think that for me that made the biting worse. I even bought my own stop-biting solution, and tried every technique I could think of, including giving myself manicures as I got older.
But as I cut back on nail biting, I substituted other "stress release" habits, like hair twisting/sucking, lip chewing, and picking and clawing at any tiny imperfection on any surface I touched, including my skin - yuk. I still bite my nails or cuticles if they develop any little snag (a sister and my daughter have this tendency, too, so there may well be an inherited tendency.) Lately I find myself yanking out my eyebrow hairs if I'm immobilized in a meeting or by illness.
I'm a fairly disciplined person in many ways. I do not want to do any of these things. I feel quite a bit of embarrassment over them, and am constantly stopping myself when I notice. But when I'm doing them, my attention is elsewhere, and I'm always feeling some combination of worried, stressed, tense and restless.
For me, the best solution is often to get up and move around, do laundry or gardening, go for a walk, anything that will get me mobile. I wonder if your daughter has a lot of pent-up energy that she must control, creating some unconscious tension, when she's doing her nibbling. Maybe she looks relaxed in front of the TV, but her body could be begging for movement.
And like you, I am concerned about what ingredients are in any sort of polish or nasty-tasting stuff you might use. I hope you'll investigate carefully. Standard nail polish contains highly toxic fumes during application, and I wouldn't want my child to nibble at the plastic after it's dry, either.
I hope you will work to reduce your anxiety about this, because your intensity could inadvertently increase your daughter's stress, if that's the root cause.
By the way, another mom asked a similar question recently and got quite a few interesting responses. You might want to check this out at http://www.mamasource.com/request/recent/97013/1202607294/85