What is the fastest, in terms of moves, for a legal game to come to a checkmate, either black or white with a Knight and such that both all other cases of the checkmated King are attacked by an opposing Knight and the King isn't on the edge of the board?
Well, for starters, you need
at least five knights, because each knight can only cover two squares out of the nine necessary (the king's current square and all adjacent ones)
Here's one rather quick way to do that:
1. d4 e5 2. dxe5 Nf6 3. exf6 Na6 4. fxg7 Ke7 5. g8=N+ Ke6 6. Nd2 Kd5 7. g4 f6 8. g5 Be7 9. gxf6 Nb8 10. fxe7 Na6 11. exd8=N Nb8 12. Nxb7 Na6 13. b4 Nb8 14. b5 Nc6 15. b6 Nd8 16. bxc7 Rb8 17. cxd8=N Ra8 18. Ngf3 Rb8 19. Nf6#
The game can be replayed here, this is the final position: