I found a Headlite charger (type HLC0001) in my spares crate. Looking inside, that particular design of charger is very simple, and if that's the type of charger you have, it isn't obvious where a charger fault could be that would cause the 4-5 seconds of LED lighting up to occur.
Mind you, it isn't immediately obvious what kind of battery fault would cause the symptoms you noted. Effectively, the way the charger *I* have in front of me works, the voltage across the battery would need to rise very high before the LED cut out, which would mean either a seriously damaged cell or a bad (high resistance) connection, either of which would very likely make the pack appear stone dead trying to run a light off it.
Does it work at all driving a light, and if not, did it die slowly when you last used it, or did it just cut out suddenly?
Had it been stored for a long time before you noticed it was dead?
If your charger is a type HLC0001, what happens if you short the output pins together for a few seconds - does the LED behave as before, or stay on as long as the short is present?