I know some of the meanings of なんて and here I think it's just a word to emphasize 病気 in the sentence, but I don't understand that もんに. I think it has something to do with もの, maybe the regular 物. Also, I'm not sure but it seems that the reading of 縁 is "えん", which apparently means either fate or relationship/link, because there's also a "ふち" reading which means edge but it wouldn't make sense here. My guess:
I usually didn't have any connection with the illness.
Also, if I was right with my guess, can I say that "もんに縁がなかった" has a literal meaning of "There wasn't a connection with that thing (もの→病気)"? In this case, is もんに repeating the noun 病気, similar to そんなものに?