In the position you sent, the White player has given up a Knight for a pawn so you're totally winning, despite his impressive center. The best second move in the King's Knight is Nc6 but Nf6 is also fine and you will meet Bc4 with Bc5 and Ng5 with castling so no one is going to capture on F7. I also recommend to play other openings as black if you really hate the King's Knight
How to defend against King knight's opening?

In the position you sent, the White player has given up a Knight for a pawn so you're totally winning, despite his impressive center. The best second move in the King's Knight is Nc6 but Nf6 is also fine and you will meet Bc4 with Bc5 and Ng5 with castling so no one is going to capture on F7. I also recommend to play other openings as black if you really hate the King's Knight
Oh, you are right. I must have done it wrong then.
I do have to play other openings, I've been doing same openings for way too long.

In the position you show white sacrificed the knight for central domination and mobility. Your best move of course will depend on white’s next move. However, odds are they will attack f7 with the queen and bishop. This could be a dangerous attack since white can also play e5 to drive the Knight from f6. Your focus should be in defending against the attack and not on how to develop. ConfusedGhoul is correct you are ahead materially, the question is can you withstand the coming attack and hold onto your material advantage?
I like 5.. d6, by this move you have two attackers on e5 which should enable you to play Nf6 shortly. It also enables you to play Be6 in response to Bc4. As for the King's Bishop odds are you will play it to e2.
Concerning other openings you may enjoy the French as it is the best way to avoid the crazy tactical openings that can occur in the King’s Knight Game.
Even a bad opening is dangrous if you do not understand it!

Basically, the opening you showed us is a sacrifice with limited payback. Don't worry about it, since it's normally already a mistake.
2...Nc6 has been considered the best response to 2.Nf3 for the last four centuries, and it is played in the overwhelming majority of grandmaster games. In your example in post #1, White's 3.Nxe5 is a sacrifice that any good player would accept with pleasure! It's true that White gets a big center for his piece--temporarily--but you have a whole piece! At the moment, you have one piece developed, and White has none. There's a lot of work to be done, but Black is clearly much better. A piece is a piece!
Pawn to F6 also doesn't work since I am weakening my king castling.
Pawn to D6 blocks my bishop, but I guess this one is not too bad, after I move pawn to G6, it will open up a spot for a bishop.