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First, I would just replace the capacitor. You'd be surprised how many bad capacitors there are in normal products, like LCD monitors. I've fixed at least 10 LCD's that wouldn't work because a few capacitors had died. Usually you can tell by looking at them, but sometimes they look fine, but still perform poorly. Plus, they're cheap, so replacing them isn't a big deal.
Then, I guess I would just replace the other components until it works, starting with the red thing (I don't know what it is). There's not too many parts, so it wouldn't cost that much to replace them, and it doesn't look like it would be too hard to resolder them either.
The LED's themselves could still be bad. Maybe they just require a higher voltage now than they did when it was first made, so when you discharged the cap, it let out enough voltage to power the LED's for a second. If it turns out the LED's are bad, it's probably not worth replacing them - it looks like they're each wrapped in that palstic, and it doesn't look like it would come off easily. If it does though, then they could be replaced as well.
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