Classic, drawn game, no matter who has the move. White's only hope is to secure the queening tile by landing his king in the vicinity (e.g. f7, f8, d7 or d8). This, however, is entirely preventable.
6. Kf5e5
yes, yes...
6... Ke7e8
and the cycle is finished again.
7. Ke5d6
White mixes it up by going queenside - exciting - but the result is the same.
7... Ke8d8
Black also mirrors the cycle knowing he's iced the draw: 8.e7+,Ke8 when 9.Ke6 is stalemate and any other white king-move allows 9...Kxe7 draw.
8. Kd6c6
? Realising his predicament, white tries something leftfield.
8... Kd8e7
Black's no fool.
9. Kc6d5
Moves to protect his bride-to-be.
9... Ke7e8
! Black avoids the trap. ...Kd8 allows white to take the Opposition with 10.Kd6 and mate follows shortly: ...Ke8 11.e7,Kf7 12.Kd7 1/0; as it is though we have a draw: 10. Kd6, Kd8 11.e7+,Ke8 12.Ke6 stalemate