Rough, easy DIY method of measuring magnetic field strength

The easiest method is to put a magnetic compass on one of the magnet's axes of symmetry, and orient the compass and magnet such that the magnet's field is perpendicular to the earth's. Then the tangent of the deflection angle is equal to the ratio of the fields.

For what it's worth: - a home-made device based on a Hall effect device - for about $40.

The easiest way to make a suitable coil is experimental. After all, what you want is not a precision measurement of the magnet, but all you want is to generate the necessary electricity for your LED and you don't need any math for that. Simply make a coil that satisfies your geometry out of some wire with reasonable resistance. In this case, I would estimate that a coil with around 1-10Ohm resistance will probably get you there. Add turns and see if the LED gets visually brighter. Keep adding until you are happy with the result. If the larger coil produces an inferior result, take some turns away until you are happy...

The last time I did something like this, my intuition got me within 10% of the required result... without any calculation. And how do you get an intuition like that? By grabbing a piece of wire and trying until what you want works the way you want it! After some time you will get quite good at estimating how to go about these things.

Now... that's not what an engineer will do, but that's pretty much what an experimental physicist will do for a non-critical application like this.