1. The schematic is hopefully legible for you and is written on my "white board"
2. The breadboard circuit as shown.
3. The 2 coils for comparative sizing. The coils were not particularly tightly wound and I didn't take any special care when making them perfectly round or having a matched Q.
4. My scope showing th output of the 555 timer producing about 138khz.
5. Me holding the receiver circuit. I did wrap it in electrical tape and added some hot glue just to keep it from deforming and unwinding itself. I was able to use a few different LED's as the "load" including a white or blue LED. I didn't try any other type of load.
Please try the schematic as shown and the coils as provided and hopefully it will work out well for you. One last note the FET can become a bit hot without a heatsink so I usually only leave it plugged in for a few seconds before disconnecting it. Heat sinking or choosing an appropriately large driver could solve this problem. Play with it and let me know what you find.