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I don’t know how this fits in with latest wisdom in teaching kids but the Yamaha Guitalele could be just the ticket? It’s a slightly bigger than a Uke sized guitar. Six strings, tuned to standard tuning but a perfect fourth up. Like a guitar with a capo on the 5th fret. I have one for holidays and idle strumming. It has proper tuners, plays great, and sounds good. It’s tough enough that I leave it lying about for the kids to bash away at. I honestly love mine. Written loads of songs on it.
I've just ordered a ukulele ☹
Re: c00ps jr. I'd recommend getting a used Baby Taylor. They're not too expensive, worth the money and if she doesn't stick with it it'll either be nice for you to mess about with or have some sort of resale value.