Hey Eliza,
Let me just say that it sounds like you’re doing some great work with her! Her head voice is sounding nice and sweet and there’s some really good stuff going on.
But what I can hear is a lot of tension in a few places. First of all, the nasal sound we’re hearing is caused by tension in the pharynx, soft palate and tongue root. That shortens the amount of available space in the back of the throat and causes that harsh, piercing sound. Tension in the tensor palatini and palatoglossus muscle can also prevent the soft palate from rising, causing a nasal sound.
All this could come from two places. The first, as Kat says, is over-singing – trying to ‘micromanage’ the sound into something nice can cause tension in all sorts of areas. I have a great cure for this – get her to sing as though she’s pissed off, using a ‘blah-blah-blah’ or ‘yeah-yeah-yeah’ as a reminder. If you aim it in the right way, she will feel a releasing feeling in the tongue and jaw. I also use the ‘me-nah-me-nah’ to get this release, as in here: http://mattpocock.com/2015/10/30/finding-helens-chest-voice-episode-2/.
From there, you can build the singer on top of that, getting her to make things nice while still feeling the physical release she gets when she’s pissed off. I call that relaxation ‘base camp’, then all the nice sounds are little ways up the mountain.
The second thing could be muscle tension dysphonia. That’s when there is laryngeal tension in her natural speaking voice which prevents her making a good tone. This is worth a google, since it might tell you if it’s worth sending her to a Speech & Language Therapist. But as a halfway house, I use this exercise to release laryngeal tension: http://mattpocock.com/2015/10/29/inside-matts-studio-the-ultimate-vocal-warm-down/
Hope that helps! But it sounds like you’re on the right track with her already, just making her aware of when she’s tensing is the next big step!