Hey Rollie,
Welcome to my world. To anyone who doesn't know, I teach middle school choir, and it is messed up system where students HAVE to take a music class (very few exceptions, exceptions that are usually based on special education needs). We don't have general music, so students sign up for band, choir, or orchestra. It isn't an exaggeration to say that orchestra draws many of our best and brightest, and that both orchestra and band require more practice outside of school, and have a higher cost (instruments, supplies, and method books).
Therefore choir is made up of kids that want to sing (and I get kids that leave orchestra and band TO sing), and those that aren't in instrumental music. Numerically, I have well over 350 students, band has 140, and orchestra about 55. I'd say that if students didn't have to be in music, we would see those numbers drop to 200, 90, and 45. That means I am teaching at least 150 students that would rather be anywhere else.
Sadly, the mix between the climate of our building and the general trend for the current generation in secondary schools means that students simply won't participate if they don't want to. No one can make them--not me, not administration, and not parents (but, to be honest, there is usually a correlation between parent involvement and encouragement and participation).
So...I am left trying to make things work with choirs that have large numbers of kids that won't sing. Some of that is age appropriate--singing is one of the most intensely personal things you can do. You can quit trumpet if you are bad at it, and blame the trumpet. You can't rip your vocal folds out of your neck and blame your vocal folds. You are stuck with the voice that you have. So there is embarassment, puberty, al of those things combined. Added to the problem is the number of girls that sing at a male's range--thanks to popular music and tenors (I am a tenor) that sing too high for the general population. What we end up with is a large group of female students who can sing along with the radio, but avoid singing at their correct octave. Singing is a muscular function--and if you don't train the normal range of the singing voice, it is breathy and not pleasing to hear--and thus if you keep avoiding it, it doesn't get any better.
So, what do I do?
First, I added ukulele, thinking that some exposure to the instrument might give some of these students an opportunity to do something musical in place of singing. In reality, the kids that do nothing continue to do nothing. My guess this is true for those kids in anything that they are not passionate about. I have kids that complain, "I thought we were supposed to SING in this class, " but when we do sing, they don't. What the message really is: "I won't do anything I don't want to do."
I wasn't that kid growing up, and neither were my friends. I didn't see a lot of the other kids doing that when I grew up. I would guess that perhaps 10-20% of kids were like that when I was in school. Today is it more like 50-60%. So it is a generational shift.
Second, I try to choose music they will sing. We sing non-religious holiday tunes for our Holiday Concert, and "safe" pop tunes for the Spring Concert. For our ukulele sessions, I am basing all of our work on pop-music play-Alongs on YouTube, such as those by Dr. Jill Reese, Dr. A, and Kevin Way. For the Spring Concert, I ask kids to suggest songs and choose some of them. For this concert, choirs are singing songs such as "I Don't Know My Name," "Clouds," "I Can't Help Falling in Love with You," and "House of Gold." I throw in some folk and classical songs, too, but let them choose which of the songs we learned to perform for the concert. On a positive note, even the folk and classical songs had students that wanted to learn them.
Anyway--choosing the music that appeals to them and lets them have SOME say at this age level works a little bit (at the high school level, you can usually just choose the music--but student choice somewhere in the year is still a good thing). In the end, I look around, and sometimes there are 5 or 6 more kids singing in class, even on a particular song--and that is a win. With the video play-Alongs, it is fun to see a number of them singing along--but it is really tricking them to do so.
So...it isn't easy, and it sounds like you are doing the right things. Don't make her perform publically unless she wants to.
Now...put this into adult culture, and people are just as reluctant to sing at ukulele jams that I have attended as my students are. Our leader encourages singing--as do I. But it all goes back to that personal nature of the voice, and the fact that someone might have told a person that they stink at singing--that is personal and it can stick FOREVER. I work hard to address problems with the voice by addressing the issue rather than labeling the voice itself as bad.
And I suppose it doesn't help sometimes to have a teacher (or fellow ukulele jam member) who isn't intimidated by singing and who just sings out no matter what.
On a similar note, churches seem awfully quiet during worship these days, too--a lot of churches have moved to rock-based worship that is almost more like a concert, where the audience watches, rather than having audiences actively participate. However--people seem to like that.
It's a big issue...one that I face daily. I'd just keep encouraging her to sing (even if she doesn't). She may be singing back home in her room when she practices. That's how you know they are hooked--when their parents come up to you and complain that they are locked in their room with their ukulele.
I just wish that was more of them!