It's likely that whatever caused your power supply cable to melt also caused some internal damage to your X3 Live. The only way to know for sure would be to take it for servicing. An inspection (which you would have to pay for) would tell you what needs to be replaced and the estimated cost.
As a diagnostic startin point I would plug in my guitar, select a patch that has Guitar as the Tone 1 input, and engage the Tuner. If the tuner does not recognize the guitar signal then there's definitely an internal problem. If the tuner works then I would use headphones to see if there's any output signal there. Finally, check all cables leading from your X3 outputs.
But my gut feeling?..... Your X3 Live has been fried and it will cost more to fix it than to get another pedalboard.
you may try reinstalling the Flash Memory, in 'safe mode' if necessary: http://line6.com/support/docs/DOC-4698
if this doesn't cure it, then contact your local Line6 Service/Distributor:
Luthman Norway AS
BOX 2174 Grünerløkka
0505 OSLO,+47 2238 4425 (phone)
+47 2238 4426 (fax)