We spent a while talking about this on chat tonight, and I think I understand a little better now thanks to Chocolate and Yang Muye. So I'm going to try to write up the conclusions I came to in an answer.
Rules for だい and かい
I think the description given in the grammar dictionaries is fairly accurate for today's Japanese, but it may be a bit of a simplification, and is perhaps not 100% descriptively accurate. After all, we know from 非回答者's comment and from the fact that 夏目漱石 wrote it that かい must be possible! The only question is how we should make sense of it.
To get started, I read the rules on the page 非回答者 linked to and went through the quiz at the bottom, and I discovered something very interesting:
I can get 100% of the answers right with two simple rules!
Rule 1. If you can't use だ, you can't use だい.
Rule 2. If you can't use か, you can't use かい.
That is, I ignored the presence of い entirely and focused on whether だ, か, or の was grammatically possible. In all cases I ended up with the right answers. You'll note that the quiz never makes you decide between だい and かい when both だ and か are possible.
And although these rules aren't quite adequate on their own, they seem natural enough if you think of い as a separate particle. In fact, dictionaries list い as a sentence-final particle. That leads us to rule #3:
Rule 3. The particle い is unambiguously clause-final.
What does this mean? Compare the following examples:
The first example sounds incomplete—you could complete it by adding something to the end, like
会議はいつか、知ってますか？. But because the second example ends with い, a sentence-final particle, it seems like a complete sentence. So this is one place that かい is possible that か would seem strange.
So this is a slight revision to our rules #1 and #2, which otherwise serve us quite well. Now, let's move on to the last two rules:
Rule 4. When there's no question word, you can't use だい.
Rule 5. When there is a question word, you usually use だい.
That is to say, だい is more common in today's Japanese than かい with a question word. And while かい is still possible today, I think it's a less common than だい when both are possible, and in some cases it might seem a little bit old-fashioned sounding. Chocolate reported in chat that it was the sort of thing you might find in a 夏目漱石 novel, and of course that's exactly what your question is about.
So I think that Makino et al. probably simplified things slightly in their grammar dictionaries. Because it's more common to use だい than かい today, their rules work fairly well most of the time, but they might not have 100% descriptive adequacy.
I've come up with an explanation for why this the case. It's a bit of a just-so story, so I wouldn't put too much stock in it, but helps everything make sense in my head, so I thought I'd include it here:
I think 誰だい is possible for the same reason you can say 誰だ—the presence of the question word 誰 makes it obvious that you're asking a question, even with だ and without an overt marker like か or rising intonation. That doesn't force you to use だい, but it makes it possible, and practically speaking it's more common to use だい than かい when both are possible syntactically.
On the other hand, a sentence like おいしいのだい fails as a question for the same reason おいしいのだ does. Without a question word, you need another way to signal to the listener that you're asking a question, and that most likely means using rising intonation or an overt question particle. This rules out だ and therefore だい as well, as they would be taken as signals that you're not asking a question.
Of course, I never use だい or かい in my own speech so these rules are somewhat academic for me :-) But I hope either the rules or the reasoning help you understand why it was okay that he used かい in his writing.