Yes, you could do it yourself; you'd have to be good at soldering and very careful. I wouldn't recommend it unless you're experienced with electronic rework.

The part number from www.farnell.com is 303-6790, it's £1.43.

No idea what would make it fail, though. It's charged fairly slowly and is rated at 5v (we only use it to about 3.5v max).

Hugo