In satipatthana, how does mindfulness lead to nirodha?
The between cessation and nirodha is "cessation" is an English word and "nirodha" is a Pali word that often does not literally mean "cessation" although often it does.   

In meditation, the important "cessations" to be experienced are: 


Cessation of craving.
Cessation of attachment. 
Cessation of I-making &amp; my-making. 
Cessation of suffering. 
Cessation of ignorance. 
Cessation of ego-birthing, ego-aging &amp; ego-dying. 


In summary, all of the above "cessations" amount to the cessation of suffering (dukkha-nirodha). 

Therefore, often, when the Pali teachings refer to "consciousness-nirodha"; "nama-rupa-nirodha"; "sense-contact-nirodha" and "feeling-nirodha"; what this means is consciousness, nama-rupa (mind-body), sense contact &amp; feeling are no longer imprisoned by ignorance, craving, egoism &amp; suffering.