Hi Per, I think hearing aids act in the same manner as maskers, amplified environmental sounds help to drown tinnitus. The problem with very high pitched tinnitus is that there usually is not very much sound in say 10-20kHz range in normal environment amplified or not. That said, I have no personal experience on hearing aids.
I've understood that pink noise is often preferred in treatment of H, because its frequency distribution is close to 'natural' or 'pleasant'. From the memory in pink noise the acoustic energy decreases 6dB per octave. White noise contains all audio frequencies at the same level and sound more 'trebley'.
I also have 'old TV'-tinnitus (some days anyway) and my audiogram (up to 12 kHz) showed asynchronous hearing loss above 10kHz.