I dunno if you're still looking for a solution for this, but I believe the LED FPC in question goes by FH33-12S-0.5SH(05). Its friends call it HFS12CT.
At least, that's what I've seen it called in a few other datasheets. Should have a 0.5mm pitch. Any ideas further ideas as to how to extend this connection? I'd sure love to use a wide gamut 15.6" panel that has the same connection. Lots of FH33-12S-0.5SH(10)'s out there, can't seem to find any (05)'s at the moment though. Not even sure what the difference is, I've basically been flailing about in the dark on this issue.
at section 5.6. Different panel, but I'm thinking the connection is the same for your 15.6" LED, I've seen that FH33-12S-0.5SH(05) mentioned somewhat regularly in the datasheets and it look superficially similar to a busted 15.6" IPS panel that I have as well (not the wide-gamut).
Here's the FH33-12-0.5SH(10): http://www.mouser.com/ProductDetail/Hirose-Connector/FH33-12S-05SH10/?qs=sGAEpiMZZMt6zyh4HYqUXPO479Dvp7h3