Maybe you've already seen these, some are 'rechargeable':
I bought a couple of black ones of these, suited my purpose with a bit of adaption (I hacked off the USB connector) ....as long as they get 5v DC from somewhere:
Unique bendable design,also has light weight. Mini LED USB Reading Light Flexible Bright Night Lamp Portable Lighting For Tablet PC Power Bank Notebook Laptop USB Flashlight. feeling so convenient. its soft shade, avoid the direct LED light shine, more effective protection of eyesight.
www.ebay.co.uk
...loads of this type of thing. Easily adapted to suit:
Mini USB LED Night Light 3/8LED 5V Bulb Lamp Gadget Reading Book Pocket Notebook. 3 led USB light Size: 60 19 9mm. 8 led USB light Size: 100 19 9mm. 24 led USB light Size: 175 19 9mm. LED Quantity: 5730 smd 3pcs or 8pcs SMD for choose.
www.ebay.co.uk