Extending LED length
You probably want solid insulated hookup wire that is about 0.5mm or 0.6mm in diameter. 



That's about AWG 22 or AWG 24 (larger AWG numbers are for thinner wire). 

As an alternative you could use bare wire and use sleeving on the wire but solid hookup wire is often cheaper and easier to get than bare wire suitable for soldering. If you get two colours (such as red and black) you can keep the polarity straight. 

It would also be a good idea to get some heat shrink tubing and put it over the LED leads where they are spliced to the wire, but be careful not to overheat the LED when you are shrinking the tubing with a hot air gun.