A Blackburn Flea bike headlight and its USB charger emerged from the packs on our Young Engineer’s Tour Easy, but the battery was completely defunct. With nothing to lose, I applied a small screwdriver to crack the case:
The battery is a single cylindrical lithium cell:
The USB charger seemed defunct, as it produced only a few dozen millivolts when connected and plugged into its wall wart. Cracking its case revealed a tiny buck power supply with no obvious damage, but also no output.
So I manually charged the cell:
Definitely not recommended practice, but a bench supply set to 4.1 V and current-limited to 100 mA gets the job done: the current stays at 100 mA while the voltage rises to 4.1 V, then the current drops to just about zero over the next few hours with cell held at 4.1 V.
Unfortunately, the cell really was defunct, even after a few cycles, so I conjured a not-dead-yet lithium cell from the heap:
Given a good supply, the Flea still works perfectly:
The yellow trace shows the battery holding at 4 V while the LED current runs at 150 mA (3 div × 50 mA/div). You wouldn’t want to run ordinary 5 mm LEDs at nearly 40 mA, but Blackburn surely specified good parts.
Replacing the Flea’s internal cell seems impossible, given its peculiar form factor, and grafting the PCB to an external cell makes no sense, given that it’d then need a custom bike mount.
So another chunk of electronics goes in the e-waste box.
Ride on!
The Smell of Molten Projects in the Morning 
just for fun, maybe you should take it into Batteries Plus
Hey, remember we live pretty close to the geometric center of nowhere in particular!
I managed to scrape away enough insulation to reveal
LIR 2477, which probably stands for “Lithium Rechargeable” size 24 mm OD × 7.7 mm thick. The BPlus site shows only a primary cell in that size and eBay has some with a three month delivery delay, but AliExpress says I’ll have a pair for six bucks in two weeks. Not that I need a tiny blinky headlight, of course.Thanks for the nudge …