Classification of all 28 Exotic 7-Spheres

Wikipedia says:

As shown by Egbert Brieskorn (1966, 1966b) (see also Hirzebruch & Mayer 1968) the intersection of the complex manifold of points in $\mathbb{C}^5$ satisfying $$a^2+b^2+c^2+d^3+e^{6k-1},$$ with a small sphere around the origin for $k = 1, 2, \dotsb, 28$ gives all $28$ possible smooth structures on the oriented $7$-sphere. Similar manifolds are called Brieskorn spheres.

Wikipedia gives three sources, but the two (linked) that may actually give the proof or expand on the above result are not in English. Are these $28$ structures the $28$ exotic spheres? Are there any known English translations or papers/books expanding on the facts above? Any insights or related articles classifying the exotic $7$-spheres would be greatly appreciated.

Yes, the 28 differentiable structures are the 28 exotic spheres. If you want a source in English, it's hard to beat the original:

Kervaire and Milnor, Groups of homotopy spheres: I, Ann. Math. (2) 77 (1963), 504-537

Their paper is also available for download here, which I believe is off of Andrew Ranicki's website.