The current tower I am stuck at in Dragonspyre is called Edrik's Vault. The reason I guessed there were at least seven floors was that I happened to die on my best attempt at the seventh floor. So I now find out in this excellent post from the Wizard 101 Central forums (Great forums I highly recommend BTW to anyone wishing to learn more about this game) that there are indeed 7 floors to this particular tower.
So I died during the last floor.
I had lost all desire to try again after that attempt. But now that I know how close I was I'm probably going to try again to solo this monstrosity. Wish me luck.