How to wire LEDs and CR2032 battery to make it work
simulate this circuit &ndash; Schematic created using CircuitLab
Figure 1. (a) What you want. (b) What you've got.
All electronic components require some voltage across them to operate. You have short-circuits (the vertical wires in Figure 1b) that connect the top and bottom of each diode. This makes a zero volt difference between the top and bottom so the LEDs can't light.
Have a look at Battery and LED without resistor where I explain some other points about this circuit.

Figure 2. The circuit from the video with one of the two conductors overlayed in black.
In Figure 2 there are two isolated conductors.

The battery negative is connected to the conductor highlighted in black.
The battery positive is connected to the gold conductor around the outside of the device.
There is no direct connection between the black and gold conductors. Having one would create a short-circuit between the coin cell terminals and discharge the coin cell very quickly.
The battery is connected between the conductors.
The four LEDs are connected between the conductors. This is equivalent to Figure 1a.
Note that the gold circuit is touching the back of the battery and the black circuit is connecting the front.