Download and 3d print the Inductive Charging Base.
You can also order one from Shapeways.
Note: This base holds the induction coil nicely in place, but it isn't necessary for charging. You can simply place the necklace on top of the bare coil and it will still work.
Cut the barrel off the end of your AC adapter and use your multimeter to determine which end is the + and which is the - wire.
Solder the + wire to the red wire and the - to the black wire on the remaining inductive coil.
Pop the coil up through the 3d printed charging base ring and then onto the cap.
Add a zip tie around the adapter wires for strain relief, and pop the bottom onto the charger. If needed, glue it all in place.