I see two problems, though there may be ways to mitigate them: you're facing the sun, therefore it's effectively in your eyes, and you'd probably lose all semblance of color balancing in the screen.
Not that the second is all that important to most people. I've even gotten used to applications like Flux[1] which change the screen's color over the day, and I find it's usually easier on my eyes. Just sayin'.
You could probably do away with the mirror, and just have a white piece of plastic. It'd be diffused by the fabric anyway, there's no need to have something mirror-like to reflect it accurately. You could possibly use the entire back panel of a laptop screen, and fold it up when not in use (brain-farts while looking at my wife's white Macbook).
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