It turned out the spinny room was solved by just holding on for longer, as a platform eventually appeared. I’d obviously just got impatient and tried something else. I got to the boss at the end of the level (Death wasn’t it, it seems), who was a big golum thing that shrank every time he was hit. Thankfully, he was really easy, as I didn’t have much health (and no energy) when I got to him.
So Level 6, and it’s a lot harder than I recall. And, if you die (as I did on the swinging chandeliers), you go right back to the beginning rather than just back to the start of the current section.