Electronic – Unable to detect Hall Effect

Your magnet has a south pole and a north pole. A noth pole will produce the opposite signal in a Hall sensor compared to a south pole and so, if you have two Hall sensors at 90 degrees, you can infer any angle from the two readings you acquire.

You have to watch out for increased noise though but this should not be a problem with reasonably sensitive devices. Check the data sheet. Worst case scenario for noise is this: -

The magnet poles are furthest away from both sensor active areas. Any rotation in either direction makes the magnetic field stronger on one sensor and if these sensors are calibrated and fairly accurate you can use the sensor that has the strongest reading to determine angle.

Having said that I'd probably go for three sensors at 120 degrees.

Minimal circuit could be one chip like an analogue asic but that would cost many, many thousands of dollars. Minimal could of course mean cheap or small or both. Stretching it a bit, minimal could mean low power.

Putting all this to one side, amplify the hall sensor's output, feed that into a suitably referenced analogue comparator. That will produce a high signal when the magnet is close. Feed that into the clock input of a D type flip flop configured as a divide by two and drive the LED, via a resistor from the output.