It did not help Mr. Kirk’s chances that only days before the election, he seemed to denigrate the background of Ms. Duckworth, 48, a military veteran who lost her legs in the Iraq war after the Black Hawk helicopter she was co-piloting was shot down by a rocket-propelled grenade in 2004. Ms. Duckworth was born in Bangkok to a Thai mother of Chinese descent, and her father was a Vietnam War veteran. She has been in the House since 2013 and worked for the Department of Veterans Affairs before that.

The re-election effort of Mr. Kirk, 57, a former Naval reservist who previously served in the House of Representatives, was complicated by a medical problem. Mr. Kirk had a stroke in January 2012, and he was sidelined for about a year during his first term in the Senate to undergo operations and rehabilitation.