Ear training is an unfortunate problem here in America. For children during their earliest formative years, precedence is given to visual and tactile learning. While this learning is undoubtedly important, too often are ears left under-developed. If hearing were trained the same way as sight, everyone would have perfect pitch.
If I were teaching this student, I would go to the most rudimentary concepts possible and begin "calibrating" their ears. I agree that the problem partly stems from listening. I believe it is also an awareness problem in addition to lack of personal aural development.
I would begin by simply having her identify sounds: cars, birds, telephone, people talking, etc. Identifying everyday sounds should be within her capabilities and should begin to boost her confidence and attitude. Having a positive attitude is crucial to learning new information.
After identifying basic, everyday sounds, I would then work on having her differentiate between "high" and "low" sounds, starting with the piano's extreme registers. I would do this both with chords and individual notes. Next, I would gradually work my way inward toward middle C with the resulting goal of her identifying higher or lower minor-seconds.
It would also be wise to have her demonstrate high and low sounds - not necessarily musical sounds, but so that she shows and awareness in registral difference.
Once she has shown an ability to differentiate between high and low sounds, I would then play a single pitch from the center of her vocal range and ask her to match that pitch. If she is incorrect, I would ask her if she sang higher or lower than the pitch I played.
Part of the problem here is that given your explanation, I believe that she has an underdeveloped awareness of her own phonating resonance. To aid in this awareness, I would have her plug her ears and attempt to match pitch while humming, using an "mmmm" vocalization. Doing this greatly amplifies perceived phonating resonance and should diminished awareness as a variable.
Once she is able to match a given pitch, I would then work to put it in context of a two note chord, working through matching each pitch of the chord. I would use a perfect-fifth as this two note chord because the perfect-fifth is the 3rd easiest interval to hear after a unison and octave respectively.
Once she is confident in matching two-note chords, I would add the corresponding 3rd to create a major triad. I would then play matching games asking her to sing different parts of the chord; obviously varying the chord each time.
This is by no means comprehensive, and should be realistically accomplished over the course of several lessons. If she is only meeting with you once a week, you, her, and her parents will see little improvement unless she is a very conscientious student.
At the least, I hope this gets the ball rolling.
Good luck, and keep us updated.