I'm at the very end of the game. Returned to the castle, entered Nystul's chamber. The finale would start when you give him Gotha's book. He should then ask you a few questions.

He doesn't ask these questions. He just says he needs some time alone with the tome, and no matter how many days I sleep through -- he never brings it up again.

Help?

This may mean that you have not completed the other two strands of the main quest, namely finding the blackrock gem in each world (and adding it to the large gem in the castle sewers) and cutting off all places of power that the Guardian uses. Nystul and Miranda will tell you respectively when you've performed the final task in each respective strand.

Do you have the djinni bottle and the horn?

As mentioned previously, there are several pre-requisites before Nystul will start asking you the questions about the guardian's spell:
* All 8 small blackrock gems must have been treated by Nystul and merged with the large blackrock gem
* All 8 of the Guardian's "lines of power" must be cut by using Altara's rod in the right location on each alternate world
* Nystul must have been supplied with Mors Gotha's Spellbook
* You must have the Horn of Praecor Loth in your possession

Once you've completed all this he should start the right conversation.

Edit: Note that to have obtained Mors Gotha's Spellbook, you need to have already bound the air daemon (Djinn) first, so essentially that is also a pre-requisite.

I wonder what happens if you never get Altara's rod because she died when you crashed the keep? Is it a dead man walking scenario then? Surely there must be a provision to allow for at least a different ending.

Is one able to get past the guardian in Killorn Keep at that point of the game?

I think it becomes a dead man walking scenario, and one way of several to get there (running out of any of several other items probably results in the same situation).

It's only a matter of defeating him, which can be done with a variety of scrolls/wands and skillful combat. The key for the door he is guarding is elsewhere in the Keep.

Did you get the horn of Praecor Loth peacefully or did you attack and kill him to get it?

I vaguely recall a bug that would prevent winning the game if you got it violently. I do not know if that was patched or not.

Actually, you reminded me of another bug that could be at play here. Binding the air daemon is very finicky. You need to throw (and break) the bottle in the centre of the Sigil of Binding (on the pentagram) otherwise it can seem to work but not all the event triggers will be switched. If it wasn't done exactly right, you can get another bottle from Zoranthus to try again.

I think the signal of a correct bottle break is that the message appears "The air daemon is absorbed [...]"

Yeah, the script is weird here. In my playthrough I first placed the bottle gently onto the pentagram, then cast Iron Skin. I wanted to strike the bottle with my sword, but accidentally picked it up instead, and the ritual still worked. Getting the message is what counts.