Choose the back side of the board in KiCad. Select Tools -> Update Board from Schematic (shortcut F8).

I found it helpful to click on each component and in the properties listed at the left (picture above), ensure the Layer is "B.Cu" (Back Copper). If somehow it is Front Copper (F.Cu), click to change it. All the parts need to be on the back copper layer to be routable.

Once all the components are on the back layer, you can position them and connect them via wire traces. 

Component Placement

First click each LED and move it to cover one of the holes in the top half of the board where there is a large star on the badge side.

Routing

The Route Tool (pointed to by the green arrow) allows you to draw copper traces between parts. Before wires connect, there are little thread wires connecting each component end. Those are called "air wires" and are a guide to show you that two parts connect.

Refer to the schematic for how things connect together.

You can move any component you want but don't move the LEDs (although you can flop them if they're upside down). They shine to the front.

As you lay out parts, you'll look to connect them via a copper trace using the Route Tool.

Place a resistor to one side of each LED, the one side that will route to the battery negative (center). Looking at the schematic, the positive of the battery holder connects to both LEDs. Route the LED to it's resistor. Join the other sides of the resistor together.

Route the common of the two resistors just tied together to leg 2 of the slide switch. Leg 1 of the switch routes to the battery negative (center) terminal.

Route the positive of the battery holder (on the side tab) to the free side of each LED.

You can see how I laid out the components above. I arranged them within the original logo boundary and left some space in the middle.

Checking the Design

Be sure there are no tiny airwires visible and all the connections are per the schematic. Then go to Inspect -> Design Rules Checker to see that all the connections are correct and there are not extra connections or unconnected items.

Note that terminal 3 of the switch is unused so no routing from it is needed. Make any corrections necessary.

Final Touches

I clicked the silkscreen layer and added a rectangle in the dimensions of a magnetic pin back of anyone wished to make this a pin or brooch.