The concept looks like you are attempting wireless transmission of electric power.
It's essential that your detection circuit be tuned to the same frequency as your transmitter. Even then I'd be surprised if a homemade magnetic field can be detected more than a foot or two away.
Perhaps a huge electromagnet would produce a field strong enough to be detected several feet away. The size used in auto salvage yards to lift automobiles.
Ideally you would find a way to make the transmitting coil produce a narrow beam of magnetic flux. Then it might create a response in your receiving coil several feet away. I'm not saying it's impossible. There may be a particular physical shape, or frequency of transmission, etc., yet to be discovered which will make it efficient.
Tesla's bifilar pancake coil has been receiving notice, as the shape of antenna which might be the key to wireless power transmission.
I am trying to sense electromagnetic field created by an iron-core coil
Magnetic field on both ends of Tx coil
There is a difference between magnetic and electromagnetic.
In fact you might have more luck doing this with electromagnetic waves (radio waves, light waves, microwaves). It involves a totally different circuit design, of course.
As you have no doubt heard, Tesla is rumored to have transmitted electrical power, to light bulbs at a distance of several miles. If the tale is true, I imagine it required enormous power to be transmitted, while a comparatively small amount was received. In other words it was probably inefficient.
If you are able to light an led at 12 feet distance, I think that will be a major accomplishment. Worthy of favorable attention from spectators.