It may be a dry joint.
A quick test to try (but doesn't mean its not if no result) is with the unit switched on gently move/jiggle the LED wires/contacts components a little to see if the movement triggers the LED to start flashing ... thats often is indicative (but not conclusive) of a dry joint issue...... This happens when the solder oxidises and basically no longer makes a good contact with the wire/component its supposedly holding together. This is common on low volume electronic items ..
If you have a soldering skills ... or a friend who does ...(With unit the power off) touch the tip of the iron to all the solder joints around the faulty component until they re-melt then remove the soldering iron quickly. the solder should reharden instantaneously but now will have a better contact. Ideally you should to remove the solder and re-solder BUT .. this is a quick fix.
Notes -
1 - ALWAYS touch the tip near to but not directly onto the soldered joint (if you do you'll just pull the solder off the joint onto the tip of the iron)
2 - don't let the tip touch for on longer than necessary to melt the solder as you may melt or damage any plastic components nearby
3 - do each joint at a time and let it cool down fully ... then onto the next.
4 - Make sure you support any wires are under stress or that might pull away ... or you'll end up with a bigger soldering workload than you planned
If all that fails then yes you might need to replace the LED .. not a hard thing BUT don't attempt if you are a novice ,,.. I'm not sure where you are located .. in the USA or UK?... in the UK we have a store called Maplin with the right gear .. and Radio Shack in the USA? (or did they go bust?) ... anyway you often find guys there that might help you out for a couple bucks or will know someone who can or teach yourself how to solder buy buying a few bits and practicing .. its not that hard ..
All that said ... its upto you .... any electronics firm or electrical repair shop will have someone that could do this ... if you're more comfortable with that i'd suggest go to them ...