You can't use a pronoun like "she" directly with a gerund/present participle like "feeling". If you want to use that word as a gerund, then you can't have the "she", and the sentence becomes "I am gong to tell her the news after feeling better". But that then would be you rather than her we are talking about. (And in any case, that expression is a bit unnatural, because it has to do with your condition rather than an event, but you can consider phrases like "after getting home" or "after eating dinner".) If you want to use "feeling" as a present participle, then you need "she is". Otherwise "after she feels better" is fine, and probably the simplest solution here.