ALL the leds (ON too) switch off
I guess the ON led is the power LED. If that goes off, you take too much current.
--> You must create a shortcut when pushing the button.
I guess it's not a NC ( Normally Close ) button, and your actual wiring is different than what your picture shows.
(Is the resistor really 1k, and 220R for the Pin13 LED ?)
Arduino has a built in LED on Pin13, you do not need that external LED, resistor and wire for a most basic button test.
And you don't need that blue capacitor (?) in your picture neither, for a first test.
Reduce it to Paul's suggestion. You can enable the builtin one with that line of code ( digitalWrite(bouton, HIGH); ), or take that 1k resistor from your picture.
And recheck this line:
const int led = 3; //la LED à la broche 13
I prefer 13 