If its one of those horrible flatflexes made with printed conductive ink on a low melting point plastic substrate, there isn't a whole lot you can do with it once multiple conductors have corroded away. You cant solder them as the conductors and substrate melt before the solder does, and extensive repairs using conductive ink are unlikely to go well even if you are experienced with a calligraphy brush. You'll also have difficulty getting the contacts flat enough.
Try cleaning the contacts as I suggested above or with *very* fine grit Wet & Dry paper, folded round the edge of a piece of foam plastic sheet so you cant apply too much pressure easily - 1200 grit would probably be suitable, you'll find it at an auto repair specialist, or with a fiberglass retractable pencil brush, set long so its soft and dragged towards the edge at an angle so the fibre tips don't dig in. Stop cleaning as soon as you get the surface tarnish off as the contacts are very thin. If it goes all the way through, as you suspect, well its FUBARed anyway, so nothing lost.
If its FUBARed, the only option (short of designing and ordering a custom flatflex or robbing spares from another unit) would be the thin wires. What's the contact pitch, what are the dimensions of the slot the flatflex contact tail passes through, what's the clearance between the board edge and the case wall with the slot, and how much clearance is there between the track side of the board and the case? We *may* be able to come up with a removable connector solution.