In this case, you should know Korean say 'recover from disease' not as '병을 낫다(나아요)'. Like wintertriangles said, the verb 낫다 doesn't have object. The correct expression is 병이 낫다. It can be understood as the disease disappears.
If you want to take the disease as the object of the verb that means to heal, it is 극복하다, 이겨내다.
e.g. 아이는 병을 이겨내요.
아이는 병을 극복했어요. (past)

Let's return to the sentence 아이는 병이 나아요.
If you find there is two subject in the sentence, you are very smart. Of course if you recognize by this sentence, it is OK.
The following is very complex thing so if you can't follow, just close this page.

It can be said there are two types of sentences that have two (or more, but they are just something like fractal) subjects linked to one verb/adjective(predicate).

First type is the sentence whose predicate can be interpreted to work for both two subjects. The sentence is a sort of this case.
아이가 나아요. The child recovers.
병이 나아요. The disease disappears.
Then you should find what belongs to another. In this case the disease belongs to the child. Then you can say '아이는 병이 나아요.'
Let's make a sentence with another example.
아이는 크다. The child is tall.
키가 크다. The height is tall.
Then the height is an attribute of the child, so you can say '아이는 키가 크다.'

Second type is the sentence whose predicate can be interpreted to work only for second subject but only indirectly for first subject. Let's feel it with an example.
토끼는 귀가 길다. The rabbit's ear is long.
The attribute - main relation is yet valid. But you can't say 토끼가 길다. (The rabbit is long.) In this case in fact you can't find a precise predicate of the first subject, rabbit. (The whole phrase after the rabbit is the predicate. Whatever,) When you are reading or listening, you can understand in the same way as you do in the first type. But to speak or write, I'm going to tell you the process that can be done.
토끼는... the rabbit ....
토끼의 귀..... the rabbit's ear ...
귀가 길다 the ear is long.
Then you can say 토끼는 귀가 길다.

If you read my article (follow the link nleconte gave), you can find the common thing among the sentences above. All have the same structure, " ...은/는 .....이/가 predicate". If you change the particles like "....이/가 .....은/는 predicate", there's now comparative meaning to the second noun.
e.g. 토끼가 귀는 길다. The rabbit's ear is long.
This sentence can imply comparative meaning such as 'the rabbit's feet are short'.
Of course you can say 토끼는 귀는 길다. or 토끼가 귀가 길다. But the first one is quite, somewhat, eccentric and the second doesn't have any focus.

cf. That 'additional insight' is about the difference between 은는/이가. I will construct how to differenciate object, subject and adverbal(?) clause.

It just depends on the verb. 나다 and 있다 and 없다 for example don't have objects, so the noun gets a subject particle. Because of this, the main noun would get a topic particle to make everything clear. 저는 강아지가 없어요. 동규는 사과가 있어요.

Kimmski gave some additional insight on these difficult distinctions.

Yes i would also like to know if anyone is there to help