[mys_721tx]

Bleach breaks down to chlorine, which in turn reacts with ammonia to form chloramine. Mustard gas has sulfur and cannot be synthesized this way.

[AnotherGoodName]

Chlorine based bleach is just one of the things that has the label "Bleach" in your household cleaning aisle.

https://en.wikipedia.org/wiki/Bleach#Classes_of_bleaches

[mys_721tx]

The only sulfur containing bleaches in that list are potassium persulfate and sodium hydrosulfite. Neither contains chlorine.

Furthermore, the synthesis of mustard gas is non-trivial. Those two chemicals are not enough to put the sulfur on a carbon chain.