Hhmm, it's hard to say what's going wrong. The Cut Path operation, or using node editing to break the path, should work. Neither one of those automatically removes the parts. You still have to select and delete them. Could that be where you got stuck?
I'll outline these for you, and maybe you can see where you went wrong.
1 -- Draw the spiral.
2 -- Draw whatever shape you want to cut out of the spiral, and move it on top of the spiral.
3 -- Select both spiral and the other shape.
4 -- Path menu > Cut Path (It might look like nothing happened, but really, the paths have been cut.)
5 -- Select the pieces that were cut, and delete them.
That's probably the fastest way to do it. But I'll quickly cover the more "manual" way too.
1 -- Draw the spiral.
2 -- Add the nodes at the places where you want to cut the path.
3 -- Using the Node tool, select all those new nodes.
4 -- Break Path at Selected Nodes button on the control bar.
5 -- Path menu > Break Apart (It might not look like anything has happened, but it did.)
6 -- Select the parts that you want to remove, and delete them.
Note that both of these processes probably result in the spiral not being a single object anymore. Probably it now is a lot of individual paths. So if you need it to be a single object, you'll need to either Group them, or use Path menu > Combine.
Does that help?