Well the other thing to try is to get the wiring a lot neater. The wires are where the signals radiate from so short and close, maybe even running together with a ground wire. If you go for a screened cable then make sure the screen is only connected to ground at one end. Make the LED lead shorter as well. Anything to reduce the radiation.
It won't be one thing but a combination of measures. Try and move the receiver antenna further away as well.
These sort of things are not easy to fix, it is more of an art, but now you know where the problem is comming from that is half the battle.