Find the two LEDs you've coded to be the dragon's eyes. Mount them inside the head using hot glue. The other two LEDs will live in the dragon's throat, providing an indirect fiery flicker inside the mouth. Hot glue them in place somewhere around the dragon's tonsil area so they don't wiggle around.
You can cut a slit in the neck and bury the wires inside, or just tape them down to the underside of the dragon's body. Use tape to hide and secure all the wires that go to both of the LED strips.