Honestly, making D&D encompass anime style gameplay would require about three changes.
1: Lightly armored fighters would need to be viable. Possibly even unarmored fighters, but I think that putting a wandering ronin in a chain shirt is an acceptable amount of D&D-ization of an anime trope. So lightly armored would probably do it.
2: Make it easier for otherwise martial characters to get their hands on one or two choice spells.
3: Make it possible for characters to obtain, as innate abilities, effects that are presently available through magical gear.
4: Make available a character class that can fly and shoot light rays at its enemies. A magical girl is mandatory.
Number 1 may be already happening.
Number 2 is mandatory. D&D would need to make it possible for a character who fights primarily with a sword to cast spells like a fire based ray attack, or a spell that lets him run on water, or any number of other spells that will probably be conveniently written up in the wizard's section of the book. Fortunately, these fighters wouldn't need many of these. Spells and abilities in anime tend to be small in number, and trademarks of the character.
Number 3 is also mandatory. Take a character who hurls a bladed weapon at an enemy such that it ricochets around and returns to the thrower. In D&D he has a weapon with the Returning property. In anime he's just that darned skilled that he can bounce the weapon back to himself and catch it. This is easy to work into a character class, since the ability is already there.
4: Warlock, anyone? Perfect magical girl. Rethemes instantly.
The components of an anime campaign are already in D&D. Most of them have been for some time. A lot of them were present way back in my well worn copy of the Rules Cyclopedia, and have only improved since then.