Getting rid of Roskam-IL-6,Biggert-IL-13,and Manzullo-IL-16 and protecting Bean-IL-8,whoever wins IL-10,Halverson-IL-11, and Foster-IL-14.
1)We place Foster-IL-14 and Manzullo-IL-16 in the same district with most of the voters from Foster's old district. It will include all of Kane County,Kendell County,DeKalb,Lee,Ogle-including Manzullo's home,and western part of DuPage County. The new IL-16 will include most of the old IL-16 plus portions of Rock Island County-IL-17. The Bureau,Henry,and Whiteside portion of the old IL-14 will go to IL-17.
2)We carve up Biggert- IL-13- place Will County in IL-11(Halverson-D),Cook County in IL-3(Lipinski-D),and DuPage County in IL-6(Roskam-R). (Biggert-IL-13) and (Roskam-IL-6) will run in the same district. The new IL-6 will gain the western portions of IL-5(Quigley-D) and IL-9(Shakowsky-D)-It will be a safe DEM district. IL-9 will gain the Eastern portion of IL-5 will gain the northern part of IL-4. IL-4(Guitteriez-D) will gain the western part of IL-7 and northern part of IL-3. IL-7 will take the northern portions of IL-1, IL-1 will take the northern portion of IL-2 and IL-2 will will take in Will and Kendell. a competive primary can help Democrats defeat Roskam-R.
3)IL-19(Shimkus-R) district gets eliminated. Springfield portion goes to IL-17, St Louis portion goes to IL-12. rest goes to IL-15.
You haven't paid attention to VRA-required minority districts. IL-7 cannot afford to lose its western end to IL-4 since that reduces the Black vote in IL-7 and the Hispanic vote in IL-4. If IL-7 compensates by moving into IL-1 and 2 too far, then it is very hard to keep both those districts majority Black. In any case neither IL-1 or IL-2 can go anywhere near Kendall without antirely sacrificing their Black majority.
IL-7 will stay the same in the west and add just enough from IL-1 to add the necessary population. IL-1 moves into IL-3 in far SW Cook and IL-2 may stretch down to eastern Kankakee to add the needed population. There aren't many options to maintain three Black-majority seats.
The easiest move for IL-4 is into IL-3's Chicago and near Cook suburban area. Projections indicate that combining the northern part of IL-4 with parts of IL-5 and IL-6 will permit the formation of a second Hispanic-majority seat. Again this constrains a lot of the moves you suggest.