Hi Sonja,
I'm so sorry you have to cull your poult. I'm not THAT experienced, but I saw that no one else had replied yet, so I'll give it a whirl.
We have only had to cull one meat chick so far, and my dbf did it by stretching the neck. Body held firmly with one hand, head held with the other hand, and pull until you feel/hear the vertibrae cracking (be careful not to pull the head completely off). The poult may shudder a bit, but that's just nerve impulses.
I have also read about people using sharp kitchen shears to remove the head quickly.
Good job doing what is best for your poult. Good luck and be strong.