If all you have is a King and a Knight, I don't think you've got enough
material to mate a solitary King using traditional chess rules, even if
your opponent were to help you out and move his King where you want it to
be moved.
However, if you are playing chess on a nonstandard board, it might be
possible. For instance, if the traditional 8x8 board had extra corners
that were squares that a King could stumble into, a 'mate' of some kind
might be possible. Naturally, we'd be talking about a pretty strange
chessboard - for instance, a non-euclidean board - perhaps with an extra
square just beyond each corner square - reachable by King or Knight, and
entered as though it were just another square on the diagonal.
But short of that, the simple answer to your question is: 'No - a King
and Knight cannot mate a solitary enemy King, on a traditional 8x8 board.,
with no other pieces to assist.'
