You are not being childish. You needed J's support and she let you know she didn't care if you had problems.
I had a similar situation in my early 20's. L and I were best friends from the time we were 7 years old...the kind of friends whose parents treat the other child like a surrogate daughter. We did everything together.
As we got older, and our lives moved in different directions, she found some other friends and basically stopped caring if I was included in whatever she was doing. It hurt, but I didn't want to give up on such a longtime friendship. We just had less and less contact. Finally, one day I saw her in a supermarket. She made a remark about the fact that I had stopped calling her, saying " you can't pick up a phone?". Earlier, I might have made an excuse, but this time I just said "I guess I can't" and walked away, leaving her with her mouth hanging open.
I didn't speak to her for 12 years, although I was not that upset any more, since I had made new friends and moved on. A couple of years ago, my sister was diagnosed with Stage 4 lung cancer and passed away 5 months later. L found out somehow, and called one of my friends, who called me and told me that L was very upset about the news. He gave me her phone number.
I called her, and we had a nice talk, but knowing the type of person she is, I still don't want to renew the friendship.
You have to do what makes you happy and comfortable. J showed her true colors when you needed her. Don't feel guilty. If you feel you need closure, you can send her a card wishing her well and leave it at that. There's no need to suggest renewing the friendship.