Ignoring verisimilitude (and for the sake of the cat girls), let's presume that a fighter with this feat could fly. To make it a little more reasonable I'd fix it thus:
Whirlybird (Fighter) Feat
you can swing your ax so fast you fly like a bird
Prerequisites: Two Weapon Fighting, Weapon Focus (Orc double ax), Str 20, Dex 15
Benefit: When using the full attack action with an orc double ax, you sacrifice your attacks to fly 10 ft. with poor maneuverability. If you use your ax for any other purpose other than flying while in flight, you immediate begin to fall. You also begin to fall if for any reason you cannot swing your ax. If you can act, you may regain flight you have lost in the following round. You take falling damage normally.
Normal: You cannot fly
Special: A fighter can select this feat as one of her fighter bonus feats.
For example, if you are dazed, you would start to fall but could regain flight if that is all you did. I would drop maneuverability to poor. It's kinda hard to steer while swinging an ax that's holding you aloft.
I'd even go so far as to demand that the haft of the ax would have to be made of darkwood or some other light material.
Still goofy if you ask me.