First I have a period video that was done less than ten years after the battle.

Part One of a Naval Training film from 1950 based upon the 1948 document "NAVPERS 91067, The Battle of Midway Including the Aleutian Phase, June 3 to June 14, 1942: Strategical and Tactical Analysis by Richard W Bates" (later Rear Admiral Bates) at the Naval War college.

I have never been able to find part two of this training film.

Warning!!
Period prejudices towards the Japanese are still very much evident in this 1950 program.  

Battle of Midway:

Video Source: https://www.youtube.com/watch?v=3HgpLzp8TGU

https://www.youtube.com/watch?v=3HgpLzp8TGU
1942 ARC Identifier 13196 / Local Identifier 80-MN-9168D.
Department of the Navy. Office of the Chief of Naval Operations.
Naval Observatory. (1942 - 09/18/1947)
Made possible by a donation from John and Paige Curran.

More reading on the subject of Midway can be found here:
https://www.history.navy.mil/content/history/nhhc/search.html?q=Midway&start=0&tag=midway

Get the NAVPERS 91067, The Battle of Midway Including the Aleutian Phase document here:
https://www.history.navy.mil/content/history/nhhc/search.html?q=Midway+richard+bates&start=0&pdf=true&ts=false

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Now a more recent retelling of the story by Montemayor. I am not exaggerating when I say that in my opinion his videos rival several commercial documentaries that have been professionally done. I thought I had posted Montemayor's video on Midway from the Japanese perspective here last year but I must have forgotten.

After an almost 15 month delay Part Two, which is now Parts Two and Three, of the Midway Battle are out.

The Battle of Midway 1942: Told from the Japanese Perspective (1/3)

Link to video: https://youtu.be/Bd8_vO5zrjo

The Battle of Midway: Hiryu’s Counterstrike (2/3)

Link to video: https://youtu.be/BXjydKPcX60

The Battle of Midway: The American Perspective and The Strategic Consequences of the Battle (3/3)

Link to video: https://youtu.be/WHO6xrSF7Sw

Montemayor has several other really good documentaries at his YouTube channel.
Montemayor: https://www.youtube.com/channel/UCX7katl3DVmch4D7LSvqbVQ

Wheels

Funny you should post this now, I watched Montemayor's documentaries two days ago. His work is exceptional, these videos are true gems. 

Those Montemayor videos are legendary.  I love the ones on the Coral Sea and Guadalcanal.