It's certainly the case that $\mathbb{R}^2\setminus J$ is path connected. So any two points in $D\setminus J$ are joined by a path in $\mathbb{R}^2$ missing $J$. If this path isn't in $D$ it hits the boundary of $J$ but then you can replace part of the path by an arc od the boundary of $D$.

It's certainly the case that $\mathbb{R}^2\setminus J$ is path connected. So any two points in $D\setminus J$ are joined by a path in $\mathbb{R}^2$ missing $J$. If this path isn't in $D$ it hits the boundary of $J$ but then replace part of the path by an arc of the boundary of $D$. Then one can replace this part of the arc by an arc just inside the boundary (there is $\epsilon>0$ such that any closed arc on the boundary not containing $J(0)$ has distance $>\epsilon$ from $J$).