Black has a king and eleven seconds. White has a king, a knight and half an hour. Neither of them can win this.
A lone king cannot win because a king cannot give check. To attack the enemy king it would have to stand on a square next to him, and that square is attacked by definition, so the two kings can never touch. No check, no checkmate, no win. It does not matter how much time is on your clock or how badly your opponent plays.
What a bare king can still do
Quite a lot, actually, and this is the part worth learning.
It can be stalemated, which is a draw. It can run to the corner and count to fifty under the fifty-move rule while an opponent who never learned the bishop and knight mate flails at it. It can be drawn by the material rules: the moment the other side is down to a king and a bishop, or a king and a knight, the game is over as insufficient material.
So a bare king is not a resignation. It is a defensive task with a half point at the end of it, and I have seen strong juniors throw those half points away in blitz because they assumed the game was already lost.
A flag fall does not turn it into a win
The rule is Article 6.9. Normally your opponent’s flag falling loses him the game, but not when you could not mate him by any possible series of legal moves. With only a king you obviously could not, so the result is ½-½.
That cuts both ways. Your flag falls while your opponent has a king and a knight: draw. Your flag falls while he has a king and a rook: you lose, because the mate exists. A pawn counts as enough material too, since it can promote. The clock side of the rule is on what happens when your time runs out.
If you are the one holding the queen and the clock, the message is the opposite: nothing about that flag rule saves you from having to know the finish. Go and drill the basic endgames until mating with rook and king is automatic, because the bare king across the table is playing for a draw and it is a draw he can actually get.