When I did my test, the ratsnest lines remained between J4 and J16. I think there were only 2. So, after putting J16 back on the PCB, those 2 traces would need to be routed manually. Another more complex trick (I have not tested this), would be to connect ONLY the J4 wires to that temporary duplicate of J16 (using breadboard view). Then autoroute should handle those wires on PCB view. To cleanup, use “Delete minus” on the temporary duplicate, move the real J16 back where it belongs, and move/drag the ends of the traces from J14 so that they actually attach (instead of just looking attached) to the J16 connector pads. This process will need cleanup on the breadboard view as well, to get rid of the dangling wires that were hooked to the duplicate of J16.
Above is not going to work.
After a closer look at breadboard view, deleting all traces from PCB view, and checking some connections on both PCB and breadboard view, I agree with Peter. The ratsnest wires on PCB view do not match the wires on breadboard view. Deleting all traces on pcb (routing menu, select all traces, delete), then doing the same on breadboard view (select all wires), followed by re-adding the wires in breadboard view, and finally autorouting pcb MAY work.
I do not know if autrouting will work after than. The problem is that EVERY single pin of J16 connects to a pin of J4, so 12 paths need to be found from J4 to J16, while still having room to connect the 12 J15 pins where they belong. There does not seem to be enough physical space on the pcb do do that. At least with a 2 layer board, which is all the Fritzing can handle. Maybe if you change the traces width to a very small size. Which is not a good idea when they need to carry power, not just digital signals.
EDIT2: I see that some of the common (grounds?) are connected in schematic view. Delete those along with the rest.