One of the things I recommend for anyone updating an operating system is to get rid of as much residual cache as possible without reinstalling. Since you are still stock everything I'm not exactly sure of the procedure, but try shutting off your phone, then hold the volume down button while pressing the power button until you get to fastboot.
It will look like this ....
Use the volume keys to cycle through the options. When it says "RECOVERY MODE" press the power button.
You will get this ...
Again, use the volume keys to highlight the last item "wipe cache partition" and press the power button to select. Make sure you don't select factory data reset. It will wipe your user data from your phone and you'll have to set it up again.
Once you've wiped the cache, reboot the phone, which may take a little longer than normal the first time you boot after wiping the cache, and see if that clears up your bluetooth issues. If not, then instead of flashing your phone back to JB, I'd try a factory data reset
AFTER BACKING UP YOUR IMPORTANT DATA!!!!. As a matter of fact, why not do that now, just to be safe.
If that doesn't work, I'd just flash the latest stock image of KitKat and repair all of your bluetooth devices. You'd have to do that anyway with JB and if you did flash to JB, you'd just get the OTA again.
Good luck and let us know how you make out.