AT&T Illinois has agreed to pay a $23 million fine to resolve a federal probe into its illegal efforts to influence former Illinois House Speaker Michael Madigan, who also faces new charges for his alleged role in the scheme, prosecutors announced Friday.Under the agreement, the company admits that it arranged to make payments to an associate of Madigan, who was one of the state’s most powerful political figures at the time, in exchange for Madigan’s help in pushing through legislation favorable to the …