What is the root of this quartic?

Of the four possible roots, I already know the approximation of the one I am interested in:

     ≈0.1729836581545961.

I need the symbolic (exact) expression for this root, probably containing square roots of square roots of rational expressions, something like:

     √(105√(167/8)

I get a couple steps past the depressed quartic in Ferrari's solution, but I have just never worked through this kind of exercise before, and I get lost.

So here is my depressed quartic:

     x^4 - 17x³ + (167/8)x² - (105/16)x + 153/256

In Excel, plugging in 0.1729836581545961 for x gives exactly 0. This is the only root I am interested in.

Thanks!