Welcome to GD.SE, and thank you for a great question. It's one that challenges designers often enough that I'm guessing there will be plenty of answers.
Based on what you say, I think the key is, "I'm confused because they are." The way to reduce any confusion, in design or anywhere else, is to pick ONE item out of the morass and start with that. It almost doesn't matter what you start with, because at this stage you need to reduce the noise level as a first step toward breaking the impasse.
Let's take the mission statement. It's probably much too wordy and a bit vague, but somewhere in there is a sentence or a phrase or a key word that you can relate to something in the real world -- an object, a piece of equipment, a type of person. It has to be something concrete. Your brief is probably 90% abstract and philosophical, but only philosophers buy philosophy, and even they respond better to a bust of Socrates than a vague abstraction.
Make a sketch or a quick mock-up of how that might look, then test it against the other parts of the brief. Does it clash with anything important? Can you think off-hand of three or four different ways to use it in layouts, ads, copy? Does it look too much like the competition? If the answers are "No," "Yes" and "No" respectively, expand your mockup into a mood board. If not, repeat the exercise. If the phrase you picked is a dead end, pull something else from the brief and try again. Keep at this until you have three possibilities that you can see have some "legs" -- simple enough and flexible enough to riff on, strong enough to be easily recognizable.
You want to end up with at least two, preferably three mood boards that you can go over with the client. Out of that meeting you will accomplish at least two things: a) you'll get a direction to start in for design development or you'll find out where not to go -- either way you'll find out where the ballpark is and whether you landed in it, and b) you'll have greatly clarified the client's thinking for them, which will make them happy and easier to work with going forward.
If the whole brief is just too self-contradictory or vaporous or incomprehensible to get any kind of corner on, you might have to highlight all the bits you don't understand, go back to the client and get them clarified. You'll be helping them as much as yourself. They need to be clear about what they're doing, otherwise you and they will spend weeks going in ever-decreasing circles. A key question that often helps in this situation is: "What makes you different?" Push for an answer that isn't something any of their competitors could say. That differentiator is one of the most important keys to effective branding.
