How do I fix this lamp?

I have a beautiful vintage art lamp that has three "arms", each bent and curved to resemble a music symbol but each arm has a lamp that is draped with a glass shade. The lamp fixtures are apparently connected to wires through each arm and come together at the bottom. When a bulb on one of the three arms burnt out recently, I unscrewed the bulb which stayed connected to the fixture and the entire thing broke off including the cord. I can't see a way to get into the sealed metal arm to replace the cord so I'm not sure what to do. Am I missing something ? I've attached pics

q fix this lamp
q fix this lamp
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