The problem is tar (the brown). It hangs onto it. And it takes a solvent to remove it. Finding a safe solvent that you can use liberally and not damage plastic/electronics (smearing conductive material across connectors) is difficult. Some of the Orange/Citrus oil products might help, but don't know if they are safe.
OH - I forget with LCDs, but CRTs, be EXTREMELY careful. There are some high capacity caps that hold a charge forever. You can get the nickname of Sparky real quick (and engraved in stone).