Good luck getting Internet people to actually explain a proof for mere mortals. The whole point is to omit as much as possible to make yourself look smart. Then you can say that it's "obvious" or it's "left as an exercise" or something dismissive rather than just adding a few extra words to explain.

For clarity: the proof hinges on (√2 - 1) < 1 (so that k’ < k) and √2 being a square root so that you can multiply k’ by it and indeed get 2 k - k √2.

This is why it also works for √3, but doesn’t work for √5 or 1.5

Right. It can also generalize for any non-integral square root of an integer √n. Let a be the integer ⌊√n⌋, such that a < √n < a+1 -- strict inequalities, because √n is non-integral. So 0 < √n - a < 1 and the same proof works with

    k' = √n k - a k         (instead of √2 k - 1 k)
       = k(√n - a)          (so 0 < k' < k)

    √n k' = n k - a k √n    (is integral; k' ∈ K)