In your line after Ke6 comes Ke4 with a drawn position. If Ke7 then Ke5 and black is holding. The reasoning is that if it is black to move, then after Ke4 comes Ke6 and black is in zugzwang, the position is of mutual zugzwang.
B & Ps vs. N & Ps
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The right defensive idea, as Mal_Smith suggested in the first post, is to put the Black pawns onto white squares. For example: