“Les onze prochains mois” or “les onze mois suivants” would be the most usual translation for “the following 11 months” to me.

I would translate the “for” with “pendant”, “durant” or “pour” depending on context.

Another usual translation (that is pretty much literally translated from the English version you provided) :

• “Il a travaillé pendant les 11 mois qui suivent. (ex: dans cette entreprise)”

Ou un exemple au passé :

• “Il a travaillé (dans cette entreprise) pendant les 11 mois qui suivirent (la mort de sa femme).”

In general one would simply say “pour onze mois” or “pendant onze mois” – if the “following” idea is implicit from the context. You should provide more context to be sure.