IMO, there is no such thing as tone deaf. I think tone deafness is a matter of inability of the vocal muscles to function, mostly due to inactivity/atrophy. With a teenager, this is likely due to the current blandness with which a lot of youngsters speak, combined with the fact that so much of pop music is so restricted and dull in terms of melodic line that even if they "sing along" they are not using much of a pitch range, hence her current range of a fifth, about what most pop music uses.
N.B. I am not down on teenagers. Adults are also guilty of talking with a limited pitch range, especially those who grew up with rock music being the principal thing they listened to. Yes, I admit I am prejudiced against the current pop music. It's boring to me.
What to do? Somehow, if this girl is to sing, you have to provide her some experience of greater pitch range. Have you tried the standard stuff, like "sirens" and whatever sort of tricks you may have in your bag to get her into higher and lower pitches? Maybe the ghost sound Peggy suggested for vibrato would help. It has to be fun. The problem is you have her for a half hour a week and the rest of the time she is doing her usual thing. Can you get her to listen to real singers between lessons, just so she can start hearing a bigger pitch range?
P.S. Whether I would teach the "apparent" tone deaf would probably depend on what their goals were. I doubt a person who starts there would ever make a real singer - not that it's impossible, but it would just take toooooooo long.
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