Cork is great for deadening sound, ie neutralizing standing wave echoes (I use it in the control room of my studio), but it doesn’t really have any sound proofing/sound insulating properties.
Matteresses as suggested as a quick fix, it just seems from your situation that you may need quite a few.
Maybe get creative with your microphone use and placement and the amount of gain you are using to record with.
What microphones/recording device/other gear are you using for your project?
adam