No it does not sound natural. Usually that phrase is used for correcting someone's mistakes on something they already did and not something they are going to do. Since this situation involves a couples relationship the word advice sounds more natural. I tried to give her advice, but she still insisted in breaking up with him. I gave her advice... She did not take my advice... She did not listen to my advice...

I may have written this sentence like this: I tried to talk some sense into her, but she still insisted on breaking up with him. In my opinion, this is more natural

"talk some sense into someone" to help someone think about something in a reasonable way YES it is natural

I don't find anything wrong in that sentence. If some one is behaving senselessly .. why not try to talk some sense in to her? Why not give some advise that brings sanity in to her/him ? If some one has already done something .. then there is no need to convince him/her to do otherwise .... right ???

When you "try to talk some sense into someone", you are trying to convince them that a choice they are making or a direction in which they are taking their life is wrong and should be changed. In the above sentence, the speaker thinks the the woman is making a big mistake by break up with "him" - a third person in this case. We assume "him" refers to a boyfriend or husband she is about to break up with. You usually use this phrase when you think that what the person is doing is very obviously wrong. "He wanted to buy that motorcycle when he has never ridden one. I've tried to talk some sense to him, but it didn't help."