Double pointed needles 2,5 mm. With a circular needle and place markers, you can knit two socks at a time.
I used a no name cotton yarn, but I can also recommend Woll Butt Sonja.
Gauge (of my no name yarn with stst): 10 x 10 cm = 45 rows, 30 sts
You can change the length of the sock yourself (for example, if the circumference which you wish is correct, but the feet of the child are a little bit shorter – my daughter’s feet are fairly narrow, so it is possible, that you must use the pattern in a smaller size).