Thread #2 joins the other end of the switch to the smaller, brass bead (+) on both LEDs.

Thread #3 joins the larger, silver bead (-) of the LEDs to the minus of the battery holder.

Start with a knot at the switch, and work your way towards the rounded tab (plus) on the battery holder.

Tie a knot and cut.

You must cut your conductive thread before proceeding.

Secure the brass beads on both lights with several passes of conductive thread.

Work your way to the switch, fasten it with a knot, then cut the thread.

Make the last connection from the silver beads on both lights to the square tab of the battery holder.

Ensure this new thread doesn't touch the previous one.

Dab a bit of fabric glue to secure your knots; wait for it to dry.