In order to make sure the microswitches are mounted at the right spot, without having to mount the drawers first, I went to my local FabLab, and made good use of their laser cutter to make small acrylic plates with holes in the right place. This way, I can just mount the plate with one big screw (IKEA screw 100344, can be had for free at the Service Area of your local store) in one of the holes meant for mounting the rails for the drawers.
Next, I add two 5mm nylon spacers and put the microswitch on top of those, fastening the whole contraption with two 2.2mm * 25mm screws.
To mount the LED-strips, I need two tiny plates with holes in the right place, so I can mount them right beneath the rail for the drawer above. Still designing those, since my first attempt on making them left me with a pile of 4mm acrylic spacers with some very specific holes. ;)
The LED-strip for the top drawer will be stuck to the metal rail on top of the cabinet.