"I am with Jack."

Translation: Jestem z Jackiem.

[Zoromskid]

But why Jacekiem vs. Jacek?

[Jellei]

Both for the English name Jack and the Polish name Jacek (which by the way comes from Hyacinth, but is a lot more popular), the Instrumental form is "Jackiem".

One of the most basic uses of Instrumental is when English uses "with". Polish then uses "z" (together with) or just bare Instrumental (when 'with' means 'using').