Earlier quoted context omitted.
Bleach breaks down to chlorine, which in turn reacts with ammonia to form chloramine. Mustard gas has sulfur and cannot be synthesized this way.
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
Furthermore, the synthesis of mustard gas is non-trivial. Those two chemicals are not enough to put the sulfur on a carbon chain.