The Carpenter worked on behalf of other supervillains to fortify and repair their lairs.
Arkham Underworld Incident
The Carpenter appears in Arkham Underworld as a constructor who can be utilized to create items purchased from the black market, as well as advance these when they are upgraded. She can be seen at work when constructing new extensions to the hideout. It is possible to attain two Carpenters by way of in-app purchases.
Arkham City Incident
Arkham Unhinged Incident
Jenna Duffy was a member of the Wonderland Gang, Jervis Tetch's first criminal organisation. She had a run-in with Batman at one point, who told her to leave Gotham. She tried, but Hugo Strange had her captured by TYGER at a bus station. She was incarcerated in Arkham City where she was located by her old gang. Tetch used one of his mind-control chips to force her to work for him again. When Batman broke up the Wonderland Gang in Arkham City, he freed Duffy, who elected to hire herself out to the various supervillains in the city, using her skills as a carpenter to repair and fortify their lairs. She had billboards up in the city advertising her 'home repair' skills.
Shortly afterward, Two-Face gave her a place on his jury to convict the Joker of his responsibility for the creation of Arkham City. She helped Dent's 'court' convict Joker of the charges because after doing a job for him in one of his hideouts, the Joker had placed bombs in it and planned to blow it up with her still inside. In the fight after the trial ended, she threw Two-Face's gavel at the back of Mad Hatter's head.
Arkham Knight Incident
Jenna apparently has established a business called Jenna's Furniture on Miagani Island.