Jackett replaced Alan Stubbs as boss of the Championship's bottom side on 21 October, but stepped down on Monday.
BBC Radio Sheffield's Rob Staton reported that future budget issues were a factor in Jackett's decision.
"From my point of the view the project with Kenny was always a three-year plan," Stewart said.
"I was very focused on building for the future, but right out of the blue I received a call after the Leeds game [a 2-1 home defeat] from Kenny to say that he wanted to resign.
"We have parted on good terms, and whilst I am very disappointed, we won't dwell and the search for his replacement is now ongoing."
The Millers have taken just seven points from their 18 league games this season and are 11 points adrift of safety.
Paul Warne is in charge of the first team on an interim basis and Stewart confirmed the board would not hurry to appoint Jackett's replacement.
"I think we have to think long and hard. When you look at the successful managers we have had in the past, they had the combination of enthusiasm and passion, coupled with the skill to get the best out of the playing staff.
"Those are the characteristics we are looking for," he added.
Rotherham chairman Tony Stewart says manager Kenny Jackett's resignation after just five games in charge of the club came "out of the blue".