Sounds like there are a couple of things that could be causing this problem. The most common, and easiest, is that the battery could be dead. It is a pretty easy fix and we have the guide for it right over here. If that isn't you issue, it could be a problem with one of the connectors or flex cables, but you should be able to notice any physical damage to those parts. Additionally, it might be an issue with the dock connector, another common ailment with this device. The guide for replacing the dock connector is linked in that second link.
Luckily, you seem to have a fully functioning secondary device with you, so you can use parts from it to test out the broken device without having to purchase anything. I would recommend that you start by swapping out the battery in the dead device for the battery you know is working. If this solves your problem, you will know that the battery was the issue! If that doesn't do it, try the same thing with the dock connector.
Side note: Just to state the obvious, couldn't you just put the logic board from the dead iPhone 4 in the working phone and use that one? Of course, then you have a dead phone laying around just asking to be fixed, I guess.
Either way, I hope this helps! Good luck! Let us know how your repair goes.