If your compact is wobbly on the table like ours, stabilize it on a towel and with a weight of some kind, for us a tape measure and paper towels worked great.
Assess your makeup compact to find the optimal positioning for your LEDs. This Shisiedo compact has a little extra space right at the edge of the mirror, but not too much outside that. The wires will have to be just the right length to travel between each sequin without getting in the way and preventing the compact from closing. Your compact may require a bit of experimentation to find the optimal placement!
Squeeze a glob of E6000 adhesive onto a plastic bag or other protected surface, and use a toothpick to pick up and spread some onto the back of a sequin, or onto the edge of the makeup mirror, and use tweezers to position and press down the sequin. Repeat for the remaining three or four sequins. Allow to dry for at least 12 hours before proceeding (24 is better).