I would like to be able to operate the LED up to a few feet away so it's not limited to being within just an inch or two of the transmitter.
The other thing I've noticed on some of the online videos is the coil that is connected to the LED is quite large. Ideally, I'm trying to get both the LED and whatever is necessary for the receiver component to fit within a 5mm W x 7mm H cylinder. Obviously, quite small. Can this be done?
Transmission range using a magnetic field is dependant on field strength and the Q of the receive coil. Using a 5mm dia coil and achieving a range of say 500mm is extremely difficult. It MAY be possible with immense receive q factor but this would require very special tuning capability.
You may be better using microwave and RF Unfortunately, at the dimensions you mention you'd need to be at 10's of GHz for the receiver aerial to approach 1 wavelength, which you'd probably need. Maybe doable but very hard.
You do not say what power level you want or how long you want them to operate for. Battery powering may be an option.
LASER optical powering may be an option.
If you better described the application you may get a better answer.
eg IF it was possible to operate the target inside a large cube whose sides extend beyond the target then you would find wireless power transfer much easier. eg a 1 metre edge cube with wires along the edges can allow power transfer into a coil of the sie you need in many locations inside the cube.