Major Kelli Bailiff, of the Wyandotte County Sheriff's Office, said it was possible the deputies were shot with their own weapons. Police say when officers arrived they were shot at, so they returned fire.
Police told KCTV they believe the inmate, who was shackled and handcuffed, used one of the deputies' guns to shoot them.
Kansas City, Kansas, police have taken over the investigation.
Kansas Governor Jeff Colyer released a statement Friday saying he was deeply saddened to learn of the shooting, and said he and his wife Ruth will be praying for the deputies and their families.
The suspect also was shot and was operated on at the hospital. The other deputy had not been identified as of Friday evening, but was in critical condition at the University of Kansas Hospital.
Moments later, the Wyandotte County sheriff's deputies were shot by one of the inmates.
Kansas City, Kansas, police say 35-year-old Patrick Rohrer died at the Kansas Medical Center from his injuries.
"Today's awful events in Kansas City bitterly remind us once again of the dangers and sacrifices faced every day by law enforcement officers throughout our state and nation", Kansas Attorney General Derek Schmidt said.
The woman with the sword was shot after barricading herself in a shed for hours, officials said.
Police Chief Rick Smith says the officers "did what they thought was necessary".
Blair added to the Star that police were not looking for any other suspects and that the incident "is contained".
Bailiff said out of respect for family members, no names are being released.
He said the shooting suspect was critically injured in the shooting and is now in surgery.
The conditions of the deputies' are unknown.
The Olathe Police Department, Raymore Police Department in Missouri and the Jackson County Sheriff's Officer in Missouri tweeted similar heartfelt messages.