I think Zuko probley thinks of Ursa (you know, his mom)while he's sick and that's when he decides to look for her.
Possible ways Zuko could get sick:
-he just gets it for no reason
-during Lake L. he gets injured and becomes sick from it
-during Lake L. he goes swimming in the lake and gets sick from that
-he gets sick from depressen
-he becomes to obcessed with capturing the avatar he forgets to take care of his own health
-he gets sick from to much change, when he finally gives up on capturing Aang he figures out Aang is right there and he can't get him
-he could have gone inside the prision and have gotten something there
-he could have been captured when looking for Appa and got sick while in prision
-along those lines, he could be tortured while in prision and get sick from that
-he could get something from just living in the Lower Ring in general
And that is my list, add as much as you want to it.
Now my thoughts on Wikipedia...
I personaly take wiki. seriosly. I mean it has been wrong numerous times, but when it is wrong they fix it pretty fast. And also wiki. has been right a lot of times before so... yeah