Without making a PCB, you could do this using copper tape, available from many picture framers. The circuit needs to be parallel, you have described series. Use 2 pieces of tape, top one +ve, bottom one -ve. Solder all +ve legs to the top track and all -ve to the bottom track. Solder a battery snap to the tracks and add a 220 ohm (red,red,brown) resistor across the tracks, this will let all LED's light up to full brightness. We did something similar to this last year in a circle around the face of a clock.