While Wisconsin, the state which you received your speeding ticket in, is not a member of the Drivers License Compact (DLC), which requires member states to exchange moving violation convictions, this state has still decide to share ticket information with the licensing state of out of state motorists that are convicted of moving violations in WI.
If you plead guilty and pay the ticket then once you have been convicted of the 13 mph over speeding citation in the Wisconsin courts they should send this information about this ticket back to Illinois.
The drivers section of the IL Secretary of State site does not mention specifically that out of state tickets are put on your driving record. However the Illinois Administrative Code Motor Vehicle gives information on the Illinois offense table. Here it notes that all out-of-state convictions are to be considered and to be coded exactly as Illinois Vehicle Code violations with the exception of the first digit which shall be an "8."
From this Administrative Code information listed above we gather that out of state moving violations will be added to your Illinois driving record and assigned points. In Illinois, speeding violation points are between 5 and 50 depending upon how much over the speed limit you were cited for.
So you are correct that in Wisconsin Speeding 11 through 19 mph over limit is a 4 point offense. However in Illinois points would be assessed and placed on your IL driving record according to the IL points schedule. In IL speeding by 11 to 14 MPH above limit is 15 points.
As for your insurance rates, if the Wisconsin speeding violation is placed on your Illinois driving record then your insurance company could see it the next time they pull your MVR. Depending upon their rating system this out of state speeding ticket may or may not raise your rates and thus affect your payment amount.
To find out about your insurer's rating system contact your insurance agent or the IL Department of Insurance who they must file their rates with.