First of all, you are in a bad position now - having the task of writing tests for the code you did not originally create and without any changes - nightmare! Talk to your boss and explain, it is not possible to test the code without making it "testable". To make code testable you usually do some important changes;
Regarding private variables. You actually never should do that. Aiming to test private variables is the first sign that something wrong with the current design. Private variables are part of the implementation, tests should focus on behavior rather of implementation details.
Sometimes, private field are exposed to public access with some getter. I do that, but try to avoid as much as possible (mark in comments, like 'used for testing').
Since you have no possibility to change the code, I don't see possibility (I mean real possibility, not like Reflection hacks etc.) to check private variable.