Ellipse Perimeter

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I've seen lots of methods of getting an approximation of the perimeter of an ellipse, however, I was wondering if there is an exact method that exists, no matter how complex.

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Yes, such a function exists. It is called the elliptic integral. Given some basic information about the ellipse one can find an analytic form using the integral. However, there is no way to express the answer (in general) using elementary functions. Using the integrals requires the arc length formula, generally taught in a first or second year calculus course.

Edit: I should note that the formula requires a modular angle and the eccentricity of the ellipse, both of which can be found using the information you say you have... just expect to get a really complicated form if you go around using the formula with arbitrary numbers (only special cases will simplify... see the first few paragraphs of the Wikipedia page for a deeper explanation of this)