There's a few ways that I could see going with this, depending on the story.
My first reaction would be to limit the frequency response of the hearing impaired girl's POV vs. that of the hearing world. Heavy compression, a little distortion. The "reality" of listening to the world through a 50Hz-6000Hz mic attached to your ear. Not very inspiring though.
Second would reinforce @ragamesound's notion of hearing with your body and not your ears. When shopping for an electric piano long ago, dealers went into great detail about how well sampled their pianos are. One of the things I heard the most raving about was "the knock" or the sound of the wood on wood impact of keys in the higher octaves. The higher the velocity with which you strike the key the greater the knock was heard. But, another of those unintended benefits of having a piano with a headphone jack is you hear the rhythm of the physical instrument itself as its played. The squeak of a pedal, the wrinkle of the leather on the stool as the player shifts, the fingernails on the "ivory". Extrapolating that idea further into an acoustic instrument I'd think of the piano's mechanisms, creaking and punctuating the rhythm of the piece. The mechanical shift of the hammers as the player depresses the soft pedal. The piano becomes more of a percussion instrument than a melodic string instrument. In complete contrast to my first reaction, LFE content may be your friend here. Making you feel the piano more than hear it.
Thirdly, would be in a similar vein to @Ryan's suggestion. To the deaf girl, her piano is the most beautiful sound in the world. In her ear she is a virtuoso, a prodigy playing a 6'11" Bösendorfer. In contrast, the hearing have the unfortunate reality of the world around them to dirty up the soundscape, in fact she's playing a honky-tonk upright, and she's hardly a virtuoso...
Maybe it's a progression of all these ideas through the story, culminating in a finale mix of all the ideas to create a new take on an old familiar piece. Again, it all depends on the story.
On a somewhat related note, I highly recommend this TED demonstration. And if you'd like more, I can't recommend this performance / documentary enough.