"now or never" means that if you don't do something now, it will be too late. I went at it again means he tried again (in this case). He repeated something that he'd done.

I had to rewrite it to clarify what I think was being said. Here are the changes I would make. "But I have seen at least two children lying dead in (a) bed of neglect in such cases, and feeling that I must get a diagnosis now or never(,/.) I went at it again." The expression "...now or never" is commonly used to say that you need to do it now or there will not be another opportunity to do it. The "...I went at it again" would imply that whatever was done preceding this sentence is going to happen again. Hope that helps.