Why won't the LED illuminate this way?
A DC generator's output voltage maximum is proportional to the speed at
which it is cranked; possibly you need to get a faster drill, or drive
the motor shaft indirectly.  You can gear up with a rubber wheel in the drill, such as some sanding drum attachments, or from a large-radius pulley with a rubber-band belt.

It takes two or three volts to make a simple LED shine brightly; some LED
lamps are actually multiples in series, can take more than that.   If you can get a voltmeter, look at the generator output, with the LED connected, for
the spin-by-hand and the spin-with-drill setups.   A series limit resistor (1k ohms) is a good precaution against overdriving the LED while you experiment.