Q:

What do all these の's do in this sentence?

病人を気遣って何が悪いののさ

is the sentence a friend sent me. The meaning is 

"What's wrong with worrying about a sick person?".

However, what do the trailing の's do and what is さ doing here? My professor explained it briefly as being similar to ですよ but I don't understand why it is being used here.
My only guess is that the first の is being used like it would be in this example sentence:

赤いセータと青いのを買った =
  I bought a red sweater and a blue one.

and that the second one is being used as a question marker turning the さ into a colloquial form of ですか

A:

First of all, the sentence:

「病人{びょうにん}を気遣{きづか}って何{なに}が悪{わる}いののさ。」

makes little sense as is because of the 「のの」 part.  As has been stated by others, that should be a single 「の」 instead.
Second of all, your professor's comment makes no sense if I may be honest.  That is if s/he is saying that 「ののさ」 or 「のさ」 means 「ですよ」.
The 「の」 in 「悪いのさ」 is a particle expressing affirmation/declaration.
The sentence-ending particle 「さ」, when used in conjunction with an interrogative (in this case, 「何」), expresses a light kind of 
refutation. 

"What the heck's wrong with worrying about a sick person?"

By now, it should be clear (if I did an OK job here) that the 「の」 has absolutely nothing to do with the 「の」 in 「赤{あか}いセータと青{あお}いのを買{か}った 。」.　The 「の」 in that sentence is a nominalizer.  "a blue one".

