I'd first try cleaning the floppy drive heads (with rubbing alcohol on
a long q-tip type stick). The heads getting dirty can make the drive
stop reading disks, and this might be what makes the green light go
off.
This stuff is too old for most modern places that do computer repair,
but if you can find someone who does classic or vintage computer
repair, they'll know about things with floppy disk drives and may be
able to help. I'm a member of the Atlanta Historical Computing
Society, and there are several members with such knowledge, and there
are likely such people wherever you are.
You might ask on this forum:
http://www.vcfed.org/forum/forum.php