Well, Enma's flame is just as overpowered as Bermuda's. 

Also, lol, that wasn't too long to read. haha. 
I see the point. However, I kinda wanted her more of a physically strong character. That's why I shied towards the Sun flames. I also figured that, out of the three Yakuza that I gave her, it wasn't befitting the Sky flame. If anything, the Sky Flame would be a member/successor of the Yamaguchi-gumi. Because I gave her the smallest of the three fictional Yakuza I wanted her to have a prominent ability, though not following the familiar Sky focus and I wanted to create an original flame for her, revolving around the Sun, that made her Yakuza family unique. 

Basically, I was running all this stuff in my head at once and it produced this formula that spat out this character in the formally overpowered fashion. I downgraded her and gave her the hypercut, but that wasn't what the original ability was. The original ability was tossing about fiery explosive flames every which way. I gave her the hypercut to match her swordsmanship fighting style, but, like you've said, I could maybe make them more Sun like and focus around increasing physical strength for the same effect (of a super effective cut). 

Though, essentially, the flames are produced by running the Sun flame through her weapon. I haven't given them any supplementary effect beyond that except the Hypercut. Basically, lol, there isn't really anything special about them other than their enhancement attribute.