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You could possibly do it, but you'd need a fine tip soldering iron, and some 32 ga solder, which if you don't already have both costs more than a new LED, unless of course you're blowing your money on super expensive LED's.
And even if you did, mishandling the legs just the slightest will break the solder binding.
dont give up hope yet! i was able to salvage a bright blue led from an old pen i found in aparking lot. both pins were completely broken off, but i wan able to use my dremmel and a file to expose some more of the metal to solder some wire onto it. i am using a standard 25 watt well iron, and standard size rosin core solder from ratshack. just dont hold the heat to long.. its best to"cheat" when u solder this type of job and put solder directly on the iron, tho tin both connections first lightly, then all u need is to hold the 2 together and the solder flows off the iron to make an ok connection. that i used some epoxy type glue (hot goue would have worked as well, but didnt have anny, so that was the only clear glue i had) and now i have a blue led
and if it dosnt work, it was already broken right? hehhe