Have you tried to do a restore to your phone? I would do the following:
a) Backup your iPhone5 with itunes.
b) Do a restore as a new phone (I know, I know hold that thought).
c) Try to pair a bluetooth device. If you cannot see any device, then you might have a faulty Bluetooth chip.
If it works,
a) Just to a backup restore back to your phone and then repair the bluetooth device.
Let me know how it goes!