"I will let my daughter go to Germany."
Translation:Ben kızımın Almanya'ya gitmesine izin vereceğim.
First, izin has two i and no e.
Also, vermek, like "to give", has two objects -- a direct object and an indirect object.
The direct object is izin "permission".
You give that permission for your daughter to go -- so you need dative case on gitmesi, i.e. gitmesine.
(As always, adding a case suffix onto a third-person possessive suffix needs a linking -n-.)
how does the ni fit in here?