There are two apostrophe errors in the sentence:
"My cat's are healthy, but my dog's leg is broken and he'll need surgery."
The first error is the use of "cat's" instead of "cats". There is no possession indicated in the sentence, so the plural form should be used.
The second error is the use of "hell" instead of "he'll". The latter is a contraction of "he will", while "hell" is a different word altogether.