This isn't a full proof, but I expect it will get you most of the way there. Let $B_2(x)=x^2-x+\frac16$ be the 2nd Bernoulli polynomial, so the quantity that you're trying to estimate is
$$ \frac1n\sum_{i=1}^n \left(B_2(\{kx_i\}) + \frac1{12}\right)
= \frac1n\sum_{i=1}^n B_2(\{kx_i\}) + \frac1{12}.
$$
The reason I've written it this way is because there is a nice estimate for sums of Bernoulli polynomials paired against the Fejér kernel: for all $x\in\mathbb R$,
$$ \sum_{k=1}^K \left(1-\frac{k}{K+1}\right)B_2(\{kx\}) \ge -\frac1{12}. $$
This is a special case of a theorem that I first saw in a paper by Blanksby and Montgomery [1]. Hindry and I give the proof of this specific case in [2], see Proposition 3.1 and Corollary 3.2. The proof uses the elementary theory of Fourier series.

So, if you average the $B_2$ part of the sum, weighted by the Fejér kernel, and flip the sums, you get
$$
\sum_{k=1}^K \left(1-\frac{k}{K+1}\right)\frac1n\sum_{i=1}^n B_2(\{kx_i\})
=
\frac1n\sum_{i=1}^n\sum_{k=1}^K \left(1-\frac{k}{K+1}\right) B_2(\{kx_i\})
\ge -\frac1{12}.
$$
This will give you a nice lower bound for the weighted average of your sums, and it remains to use this to deduce that at least one of the terms can't be too small.

BTW, Blanksby and Montgomery used their lemma to deduce a result related to Lehmer's conjecture on a lower bound for the height (Mahler measure) of an algebrac number, and Hindry and I used it for studying a conjecture of Lang on lower bounds for the canonical height of points on elliptic curves.

[1] Blanksby, P.E., Montgomery, H.L. :Algebraic integers near the unit circle. Acta Arith. 18, 355-369 (1971)

[2] Hindry, M, Silverman, J: The canonical height and integral points on elliptic curves, Inventiones Math 93, 419-450 (1988)