You'll enter the real estate taxes for the 2nd home by clicking on the Federal Taxes tab, then Deductions & Credits, then scroll down to Your Home and select Property (Real Estate) Taxes. Then follow the prompts.
TurboTax will ask if you paid property or real estate taxes and you'll click "yes." Then you'll land on the screen that says Enter the property taxes you paid. You'll notice there are two boxes, one for "main home" and one for "additional homes or land." That's where you'll enter the taxes for the IL property.