Kodai Shoujo Dogu-chan (2009)

Both feature violence and crime. A lot of crimes. The leads cohabitate and there are many shenanigans as a result.

Both of these dramas feature a physically-powerful female protagonist who battles fiercely with a sword, while displaying a jovial and gregarious personality when not actively engaging in a sword-fighting battle.

Both feature female protagonists who are blessed with everything, including a great body, smarts, ability to sympathize with others, patience, ability to react quickly, and stamina.