It turns out that $C(K) = C(\mathbb Q) = \{\infty_+, \infty_-, (0,1), (0,-1), (1,1), (1,-1)\}$.

To see this, consider a point $P \in C(K)$ and write $\bar{P}$ for its image under the nontrivial automorphism of $K$. Then $P + \bar{P}$ is an effective divisor of degree 2 on $C$, which is defined over $\mathbb Q$ (this is what Xarles is alluding to in his comment). So for any effective rational divisor $D$ of degree 2 on $C$, the linear equivalence class of $P + \bar{P} - D$ is a rational point on $J$. As you state correctly, $J(\mathbb Q)$ is finite and is isomorphic to ${\mathbb Z}/2 \times {\mathbb Z}/52$; it is generated by differences of rational points on $C$ together with the point of order 2 corresponding to the given factorization of the polynomial in the equation, which makes it easy to enumerate all these 104 points (in Magma, say). Magma represents these points as effective divisors in Mumford representation (minus a multiple of one of the points at infinity). So we look at all these points and check if these divisors are of degree 2 with points in the support defined over $K$. The only such divisors we find are sums of two rational points. This shows that there are no ``exceptional'' $K$-points, i.e., no $K$-points with $x$-coordinate not in $\mathbb Q$. If $x(P) \in \mathbb Q$ and $P \notin C(\mathbb Q)$, then $P + \bar{P}$ is in the class of pull-backs of a point on ${\mathbb P}^1$; such points all give the same point on $J$. To deal with these, we observe that such points correspond bijectively to rational points on the quadratic twist $C^{(-1)}$. However, there are no such points, since the polynomial on the right hand side of the equation of $C$ is always positive. This finishes the proof.

The business with the exceptional points is related to the fact that the map from the symmetric square $C^{(2)}$ (there is an unfortunate clash of notations with the quadratic twist) to $J$ is not injective: there is a $\mathbb P^1$ contained in $C^{(2)}$ coming from pulling back points on $\mathbb P^1$ under the hyperelliptic double cover, and this $\mathbb P^1$ gets contracted to one point, whereas the map is injective otherwise.

Since every quadratic point on $C$ gives rise to a rational point on $C^{(2)}$, the computation above actually shows that there are the following exceptional quadratic points on $C$.
$$ (\tfrac{1\pm\sqrt{-3}}{2}, 0), (\pm\sqrt{2}, \pm 4\sqrt{2}-5),
(\pm\sqrt{2}, \mp 4\sqrt{2}+5),
(\tfrac{2\pm\sqrt{2}}{2}, \tfrac{-5\mp4\sqrt{2}}{4}),
(\tfrac{2\pm\sqrt{2}}{2}, \tfrac{5\pm4\sqrt{2}}{4}),
(-1\pm\sqrt{2}, -11\pm8\sqrt{2}), (-1\pm\sqrt{2}, 11\mp8\sqrt{2})
$$
(The signs correspond in each pair of coordinates.)