If each pair of points $x,y\in K$ lie in some connected subset $K_{{x_0}x}\subset K$, then K is connected.
Fix $x_0\in K$. By hypothesis, $\forall x\in K,\exists K_{x_{0}x}\subset K$ such that $K_{x_{0}x}$ is connected. Then, $x_0\in\displaystyle{\bigcap_{x\in K}K_{x_{0}x}}.$
Since $K_{{x_0}x}$ is connected $\forall x\in K $ and $\displaystyle{\bigcap_{x\in K}K_{x_{0}x}}\not=\varnothing$, and so by rose petal theorem $\displaystyle{\bigcup_{x\in K}K_{x_{0}x}}$ is connected. Since $K_{{x_0}x}\subset K$, then K is connected.
I am not really sure about the conclusion, so I would appreciate the feedback.
Note: Rose petal theorem: If $K_\gamma$ is connected, $\forall\gamma\in\Gamma$, and $\displaystyle{\bigcap_{\gamma\in\Gamma}K_\gamma}\not=\varnothing$, then $\displaystyle{\bigcup_{\gamma\in\Gamma}K_\gamma}$ is connected.
Or simply indirectly: Assume $K$ is not connected, so $K= U\cup V$ with disjoint non-empty open $U,V$. Pick $x\in U$, $y\in V$ accordingly and assume $x,y\in K_{xy}$. Then $U,V$ show that $K_{xy}$ is not connected ...