Why Is Black forced to give up the queen in this puzzle? [FEN &quot;1r4k1/3q3p/p2P2r1/4pNNQ/5p1P/P7/6PK/8 w - - 0 1&quot;]

I got this position as a puzzle with White to play. The correct move was 1. Ne7, followed by 1... Qxe7 and 2. dxe7. To me, it seems more logical for Black to simply move the king, e.g 2... Kf8, and then capture the knight with 3...hxg6 after White plays 3. Nxg6.
How come Qxe7 is forced here after White plays Ne7?
If black plays 1... Kf8 he will get mated. Here is one possible line given by the engine:
[FEN &quot;1r4k1/3q3p/p2P2r1/4pNNQ/5p1P/P7/6PK/8 w - - 0 1&quot;]

1. Ne7+ Kf8 2. Nxh7+ Ke8 3. Nf6+ Kf8 4. Nxd7+ Ke8 5. Nxe5 f3 6. Qxg6+ Kf8 7. Qg8#

You can answer questions like that by using an engine, for example here (click on the button &quot;Toggle local evaluation&quot; in the top right). There you can also check all other variants, which you can't figure out on your own.