Black gunk on cabinets

[Jared719]

I moved into my grandparent's house and I'm trying to clean the build-up that occurs around the cabinet handles. Nothing so far has seemed to work and the cabinets have a mirror-glossy finish so I'm scared to death to mess them up worse than they are.
I've heard everything from vinegar to lemon juice. Have tried only over-the-counter cleaners so far, which seem to leave streaks.
Please help!

[Docduck]

i would remove the handles..most are only on with screws inside of the doors...use a scrubbie (soft) and some non abrasive comet or soft scrub type of cleaner and go from there...if that doesnt do the trick those areas may be worn down and the finish needs redone