this is pretty easy, the microscope comes with an incandecent bulb which is normally powered by 2 1.5v AAA batteries, just rip these all out and replace the light with a single white LED and extend its leads up through the case using the 30awg wire..
use your heatshink/tape here to insulate the leads.
test your light with a battery and make note of which lead is the anode/cathode.
on the camera board there is a small (freaking bright) orange led, *carefully* remove it, and wire the leads from our white LED in its place, with this camera the LED is under software control, usb provides all the power. make sure these leads have plenty of slack.
be generous with the hot glue to help act as strain relief on the wires; also be careful positioning the white led so that it points generally where the camera lens points