Drop an LED face down into this gap using curved tweezers, arranging them so the green line points towards the black negative wire marked earlier.
I find it's handy to make the gap between the wires slightly smaller than 2mm, so the wires pinch the LED in place by the plastic cap. This helps to stop it sucking itself back to the soldering iron by surface tension.
Continue to add LEDs into the gap and solder them in place directly next to each other, until you have the desired length of LED line. Allow it to completely cool.
Be patient, take your time, clean up messy solder as you go. To clean up lumps, run the soldering iron along the outside edge of the wire quickly, so as to not heat it up enough to remelt the existing links, but just enough to smooth out surface irregularities. This may take a few tries but you'll get the knack of it.
Repeat this step with the other section of wire. Clip off any remaining wire flush with the last LED in the line.
Once you're all done, touch the remaining loose ends to either side of your CR2032 battery to test the connection. If any LEDs do not illuminate, make sure they're placed in the right way round, and strengthen any solder connections that look bad. Once it's all good, it should look like the photo above.