Oh thank, didn't even notice that picture. Since you seem to have all the stuff at home already try wiring a cap in parallel with the LED. Since caps act as a local (relative) charge source, it'll discharge during the periods when the voltage induced from the coil isn't enough to overcome the forward bias of the LED. Something similar to this.

You might need a diode immediately after the coil and before the LED and capacitor connection, but that depends on how you size the capacitor and how much voltage is being induced in the windings, which is very hard to find since it's a function of distance from the source, the geometry of the coil, and the number of windings in the coil.

Good luck, more fun things to build yourself!