Step 4: construct the eyeball circuit
Cut an oval of white felt. Draw the rabbit face on it (2 eyes, triangle for nose position), as in the picture. You'll sew the circuit on top of this. Cut a small square of white felt, bigger than the battery, and a hole in the middle, smaller than the battery.
Using needle nose pliers, curl up the LED leads into circles so you can sew them down. Mark or remember which is positive and which is negative. Place the LEDs with both positive leads inwards. Using conductive thread, stitch from one LED positive lead to the bottom of battery holder area. Stitch over the battery area several times to make a good contact, then up to the other LED's positive lead.
Using conductive thread and leaving a long tail, stitch several lines across the top of the hole on the small square. This will sit on top of the battery and form a pressure switch: normally the threads will sit above the felt, away from the battery, but when pushed, the conductive threads will touch the battery and complete the circuit.
Now change threads to a regular, nonconductive thread. Sew the two pieces of felt together around the circumference of the battery. Leave a little space at the top (the LEDs are at the top) so you can fit the battery in afterwards.
Finally, sew the loose conductive thread ends to the LED negative leads.
Place a battery in the holder (positive side down) and touch the switch area. If your LEDs do not light up you probably have a loose piece of thread somewhere causing a short. (This is the reason I've emphasized the polarity. In theory you could swap pos & neg, but the battery's positive terminal wraps around the sides, and the bottom of the holder is much more likely to have loose conductive thread which if the negative terminal were down, could cause a short.)