I have never heard of a power adapter, 'Melting'.
If the exaust fan had malfunctioned, then the laptop would have overheated and shut down, long before the problem occurred.
I assume the power cable had a power reducing transformer (mains to 12volt), I would think this is more likely the fault.
Is the power device still functioning?, if so take it to a local shop (or friend) who has a multimeter tester, and check the output.
If the device is faulty, contact Belkin and ask their advice.
Ask a local computer dealer about repair costs, possibly a lot cheaper than PCWorld.