Strange fault. Thinking laterally.
Do you remove the plug from the wall and not from the laptop?
If not, do so, and see if the link is still okay.
What I'm getting at is that the wire is being used as an antenna - but it's no use when powered up.
This would imply that the normal miniature antenna in the laptop is not working or being swamped with something else and needs a boost from such a length of wire.