That even mild exposures to bleach cause you to “start wheezing” suggests that your asthma is not under control, thus making you more vulnerable to the effects of this irritant. You should work with your doctor to achieve optimum asthma control. This may require the regular use of an inhaled corticosteroid. This could reduce the reactivity of your lungs to the bleach. Your description also suggests that the lining of your nose is hyperreactive. This might respond to a quickly administered nasal steroid, following exposure to bleach.