You are almost done...
Let us show that every point of S is connected to (0,0). You proved that this is true for the points lying on either of the two axes.
Now, if (x,y) is in S, either x or y is rational. If x is rational, I guess you know how to connect (x,y) to the x-axis, and hence to (0,0). And you can do the same with the y-axis if y is rational.
At that point, we are done: if M and N are in S, you can connect M to (0,0) and then (0,0) to N.