When Toby is watching the wife of a kidnapped contractor speak on TV, the weather for Iowa City is visible on the screen, showing a high temperature of 75 degrees. Since it was established during the episode that it is 365 days until the end of President Bartlet's term, that would put the likely date of the episode somewhere around January 20th. However, mid-January temperatures in Iowa City average in the 30s, with a record high temperature for January of 64F, 11 degrees cooler than the temperature shown.
The plotline for "Annabeth and the First Lady" is that the first lady will attend a Martinsville NASCAR race. This occurs the day after the State of the Union in January. There is no NASCAR race at all in January and the only one the president/first lady attends is Daytona, the weekend in the middle of February.