Here's how you can make the white parts transparent.
1 - Select the white parts. Some of them are in groups, so you'll have to use the Selection tool and the status bar, to make sure that they are ungrouped. If my memory serves, I think you already know how to do that? But if not, let me know and I'll break it down into simpler steps.
2 - After they are all ungrouped, select them all, and do Path menu > Combine
3 - Select the white parts and the black part, and do Path menu > Difference
