The lighting in the instrument panel (that's what it's called, btw) is not a LED at all but a tiny bulb. In order to reach it you will need to pull off the cluster bezel, undo the screws holding the panel in place (be careful not to drop the screws because they can be very difficult to retrieve -- use a magnetic tipped screwdriver if you can find one), remove the dials (all but AC switch), disconnect the harness for the AC switch and pull off the instrument panel bezel. The bulbs slide out easily from their sockets. They have a green silicone cover which gives it that green appearance.
There are two of these bulbs. They are JKL EWG203 bulbs (very tiny). They're pretty easy to find online. Take the silicone covering off the old bulbs, put them on the new bulbs, install the bulbs and reassemble the instrument panel.
Source: I own a 2002 corolla and have done this myself.