Batwoman from earth 11, is an alternate version of Batman in a world of reversed genders. Even if she is called Kathy Kane, most of her identity traits, sans gender, are more close to Bruce Wayne than the cousin of the Batman of earth 0.
Same as Batman, Batwoman is member of the Justice Guild, an analog of the Justice League and have a daughter, Talia Kane aka Robin.
Currently, Batwoman has been transformed into the Batwoman who Laughs, and has been causing chaos through the multiverse.
The Green Lantern
When the Green Lantern of earth 0 visited earth 11, Batwoman helped him in fight an invasion from the dark multiverse, as the walls between worlds weakened, caused by Zundernell, the Golden Lantern, who was tunning earth 11 with the Dark Multiverse. Fighting the invaders of the dark Multiverse, Batwoman was cut by a dark version of Alan Scott, maybe infecting her with the essense of the darkness.
Along with the Justice Guild, Batwoman received and helped the Wonder Woman of earth 0 on her journey through world to her home.
Justice League Incarnate
During the events of Justice League Incarnate it was revealed than Batwoman, in a similar fashion to the origin of the Batman who laughs, had a transformation triggered by the death of the Joker of her world, who poisoned her before to die and transformed Kathy in the Batwoman who Laughs, a manifestation of the corruption caused by the dark multiverse in the conventional multiverse. Under that identity, Kane tried to follow the path of the Batman who laughs and using a gauntlet to travel through dimensions, went to the mystical realm beyong the chaos where Justice Incarnate was trying to take the multiversal crack powers before Darkseid, Asmodel and Glorianna could take it. However, Kane took advange of the confusion and almost took the power of the crack. However Doctor Multiverse used her powers to destabilizise the crack, sending Batwoman to the house of Heroes in the bleed space, where the champions in there were fighting against the parademons and Darkseid's Elite, lead by Grail, along with Orion and Sturmer. Batwoman joined the melee, using his power glove against all the parties involved. Under that dangers Dino-Cop ordened the Harbinger A.I. to selfdestruct so the secrets of the multiverse wouldn't fall in villians hands.
The remains of the satellite landed on earth 7. Most of the heroes and villians, including Batwoman, survived. In someway, Batwoman managed to escape from earth 7 before Thunderer unleashed the Worldstorm on his world to clean the evil influences on it.
Unstoppable Doom Patrol
Batwoman reappeared on the Bleed, where she encounter General Immortus. In some way Batwoman took control of Danny the Street and had been tortured him from a while. Her motives are unclear.
Powers and Abilities
Batwoman posses all the training and knowledge on par with her male couterpart. As the Batwoman who Laughs, she possess a golden glove of unknown origin, capable of absorb and discharge energy blasts, and possible, facilitate dimensional travel.
Previous versions of Batwoman from different alternate reversed gender worlds had appeared in the past. In Superman 349 saw himself in a world of reversed gender characters, Batwoman among them, created by Mr. Mxyzptlk.
In Superman Batman "With a vengeance" arc, Batman found himself on a world of reversed genders again created by Mxyzptlk. The Batwoman of that universe, called Helena, identify Batman civil name, surprised to see him in that garb. When the multiverse was restored in 52, this Batwoman was the basis for the official Batwoman of earth 11.
This Batwoman reappeared in the Countdown storyline, helping the Challengers of Beyond in their quest of Ray Palmer. Her civil identity was overwrote later due the events of Flashpoint.