The Knicks also reportedly waived Samuel Dalembert on Tuesday.
The trade has been characterized as a "salary dump" for the Knicks, who have the worst winning percentage in the NBA.
Thomas, a 6-foot-8 forward, has averaged 3.4 points and 2.5 rebounds in 128 games over parts of four seasons since going undrafted in 2010.
Amundson, a 6-foot-9 forward, has averaged 3.5 points and 3.5 rebounds in eight-plus seasons after going undrafted in 2006.
Kirk, a 7-foot center, has averaged 0.8 points and 0.2 rebounds over five games this season after going undrafted in this year's draft.
New York (5-32) will face the Wizards in Washington on Wednesday.