After following a tip from Jon Norris, I think I managed to improve the sound of the flute a bit. This video here is to show how the flute looked after the repairs and after putting some waxed cord to prevent it from taking off and consequent loss of air. The splitting edge is a little blunt, but I don't know what the difference in sound would be between a blunt splitting edge and a sharp one.
I want to thank Jon Norris and Keith Jones again for their help. My next flute should be a drone and I will certainly need your help again.
Strong hug.