Ne4 looks strong. Retreating loses the bishop, Ke5 Nxc5 also wins the bishop because it's trapped. Then you have some strong passed pawns.
Chess-challenge

Ne8+ is my preferred move more threatening than Ne4+ though not much between them IMO to be honest, Black can still escape quite reasonably but at a loss.

Next time post an interactive diagram so your position can be quickly copied into another interactive diagram in anyone's reply, if they're so inclined.

shippen needs to look at the position a bit more after g6. Black is dead, dead.... g6 much better than Ne8 (which is probably still winning).
Not sure about g6 as the next move though it is a good one however - does this not help black retain the initiative by Rxf6 and then black Knight can start harrassing the white King. I would have thought g6 would be the second move once King moves to Ke5. (mind you you are probably right black is doomed whichever of those three moves are made. Ne4+, Ne8+ or g6.
It's white's turn. They can make a combination to own black :)