How to insulate a finished partition?
The problem is that the inter-room partition was built without filling the interior space, without rock wool inside. One room has already been repaired. Actually how to fix the situation with noise isolation without destroying the wall?
Maybe throw a soundproofing membrane and another layer of plasterboard on top, or just a few layers to add fireproof gypsum? There is enough space, you can take another 5-10 cm from the wall for finishing. Maybe build a new frame of high quality?
If you do not want to disassemble and remodel anything, do not increase the thickness of the wall can be on the prepared ( finish putty, primed with drying) drywall glue cork wallpaper (roll), or a sheet of cork.
Cork wallpaper (or cork sheets) is a good sound-insulating material, the embossed texture of this wallpaper not only absorbs sound, but also disperses it due to cellular texture wallpaper.
Or dismantle the plasterboard and then lay stone wool and soundproofing membrane (they work well together), then mount the plasterboard.