びょうき
illness
byouki
JMdict 1490230
病気(びょうき)sickになりましたbecame
I got sick
The pattern here is “become ~”. 病気(びょうき) (sick) takes に (to) to mark the state being entered, then なる (become) carries the change — “become sick”. The verb is in polite past なりました (became), where 〜ました ((polite past)) is the polite, finished form.