Each of(各々{おのおの}の) integers(整数(は or が)) from 1 to N(1からNまでの) is written(書かれています) on a separate piece of card(それぞれ一枚のカードに).
Then the following is the best translation.
1からNまでの各々{おのおの}の整数は、それぞれ一枚のカードに書かれています。
Your translation
一からNまでの整数のめいめいはそれぞれ別のカードの一枚に書かれた。
"整数のめいめい" is just acceptable as a translation by non-native Japanese speaker, at least you should use "めいめいの整数." "それぞれの整数" is far better.
If the N numbers (from 1 to N) are written in a card, it should be traslated into "1からNまでの整数が、一枚のカードに書かれています。"
There is no need to use "各々," "それぞれ" or "めいめい" to describe this picture. There are two substances, one is a group of Numbers and the other is a card.