The general idea is to use two helical resonators (coils). It will work at frequencies from 150kHz up to 10MHz or more. Lower frequencies need bigger coils, but the driver is probably simpler. The coils radiate electromagnetic waves too, but inefficiently, so you should chose a frequency that does not interfere. The diameter can be 150mm - 500mm, and the number of turns will self resonate. Use the first link to calculate the resonant frequency. The coils should be symmetrical and identical, and made from thicker wire to reduce resistance due to skin effect (1 or 2mm diameter), or copper tube up to 6mm. Use a former so the coils stay in shape, though this will have some effect on the resonance, as will metal benches, bench tops and the floor.
The signal is a sine wave adjusted to the resonant frequency. It can be coupled in and out using a second coil (inductive link) of a few turns, though finding the exact number of link turns to match the load to the source impedance will be more efficient. Ideally there could be two leds in parallel but reverse to work on both half cycles. Use ultra bright leds as they are much brighter for a given power.
The signal/power source.
Initially use a function generator for a sine-wave signal. With 50 ohm output impedance and the voltage set to a little below the maximum (say 14.14V p-p) the voltage when loaded with a 50 ohms resistor is half, and so 7.07V p-p. Divided by 2.818 to get rms voltage, it is 2.5V RMS. WIth 2.5V and 50 ohms the power is V^2/R = 125mW. This should light the leds, but not sure of the range - maybe 0.5m or more. A power up to 1W will give greater range. The range is not likely to be more than about 1/10 wavelength.
A 1W RF amplifier is shown in the first link. It will need a stable sine wave driving source of about 20-100mW to drive it. You can scale the values of L and C for other frequencies. An amateur radio low power transmitter may suit, at least for ideas. These are transmitters up to about 1W. The second to fourth links show examples. The crystal controlled types are not readily tunable to suit the coil resonance. The third link shows an adjustable type, that is useful for a few percent of frequency change, and the fourth link shows a type that would suit if you can figure out the Cyrillic chars.
You will need experience to adapt and make these radio circuits. If you get hold of the ARRL handbook it may help.
An altogether different approach is to use a current mode op-amp like the LT1206. This has about 20Vp-p with a 50 ohm load, so 1W, and will operate up to 15 or 20MHz at this level using +/-15V with current rating up to several hundred mA. This will need a suitable oscillator to drive it - perhaps a VFO circuit for radio amateurs