Puzzle solution : Ne7+ Kh8 Rxh7+ Nxh7 Bxc6 dxc6 Qxh7+ Kxh7 Kg2#
Great work on finding the right idea but look closely at your move order for an even better option in case black decides to give up the queen to stop mate.
Below I made a slight modification of the chess problem to illustrate my point. Checkmate can now be forced in 6 moves. Although the mating line is one move longer than the variation you gave in the original problem, there is greater clarity due to no ability of black to stop checkmate.
People who already saw your proposed solution of the original problem can still try to solve this one. The shortest forcing line will look slightly different.
In the original problem we should play Bxc6 before Rxh7+ so that if black sacrifices his queen to stop mate we can keep our rook!
Ne7+ Kh8 Bxc6 and if black does not sacrifice his queen then we can play Rxh7+ (or Qxh7+) and transpose to the mating line given by obscurebean. If he does sacrifice his queen then our material advantage will be greater because we have not played Rxh7+ yet.
In the revised problem we can play 1.Ne7+ Kh8 2.Qxh7+ (our rook on h3 is actually a better attacker than our queen on b1 after our next move) Nxh7 3.Be4 and black never gets a chance to sacrifice his queen after f5 4.Rxh7+ and 5.Kg2# after g6 then 4.Bxg6 fxg6 5.Be5+ Rf6 6.Bxf6# or 4.Bxg6 Kg7 5.Bxe5+ f6(or Nf6) 6.Rh7#
The revised problem took me a long time to figure out. Is it a coincidence that adding a Bishop to the starting position changed the problem the way it did or did you think of it beforehand but decided the first problem was better for some reason? I don't think I would ever be able to solve this if you did not tell us it was a forced mate in 6. When you said that I decided that for the f4 bishop to matter the e5 to h8 diagonal would need to be opened. Since our king still needs to move in some lines to deliver mate, Be4 instead of Bxc6 made since once I realized it would force g6 and not allow the black queen to be sacrificed to stop mate.
I actually found many ways to win in the revised problem. The hardest part for me was finding why 2.Qxh7+ would lead to a faster win that 2.Rxh7+ (easy win but longer than 6 moves).
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