with the 120 to the coil, the relay is activated.
with the relay activated, the contacts are on the N/O output
when there is a power failure, the relay changes state to the non-powered condition and the N/C normally closed terminal will have the battery power.
the LED connected to the battery and the relay.
the battery connected to the LED and relay.
if you do not have a 120V relay now, but you do have a 5V DC, then get a phone charger and power the relay.
this is how it is done in every commercial building in the USA.
required over each exit door, and every so many sq feet for emergency lighting as required by fire code.
the EXIT or EMERGENCY lights also have a battery charging circuit that is connected when there is power, but that is easy enough if you use a DPDT relay and use the relay to also isolate the battery when there is 120V power.
remember, there is no direction on a relay, you can connect one wire to either comm or NC, or to NC or comm, whichever is easier or offers you the most benifit.
with a DPDT relay, the other pole can be how you connect the charger to the battery when 120V is present.