We have labelled each part of the text with what we think is the relevant category by putting our analysis in square brackets immediately before the relevant part. We have added comments, where appropriate, after our analysis of each sentence.
[DS] 'How's Mother?' [N] she asked dutifully.
The reporting clause indicates that although what Lolita says is in DS, she is not really that interested in the question she has asked.
[N] I said [FIS] the doctors did not quite know yet what the trouble was.
We have annotated the material after the reporting clause as FIS because, although it is grammatically subordinated to the reporting clause and a number of other factors are typical if IS (e.g. the verb tense) , and so looks like IS at first sight, the adverbial 'quite' and the deictic adverb 'yet' are more appropriate in deictic and attitudinal terms to the situation where Humbert Humbert is talking directly to Lolita. Hence these forms should not appear in an IS construction.
[FIS-DS] Anyway, something abdominal.
Given the obvious links between this sentence and the previous one (they are part of the same utterance), one might expect this sentence to be FIS, like the last one, and it occurs in the same paragraph and with no quotation marks. But the fact that there are no pronouns and no verbs means that it could just as easily be a form of DS, but without quotation marks. Hence we have annotated this as ambiguous between IS and FIS.
[FIS-DS] Abominable?
The same sorts of argument apply here as for the previous sentence, even though this sentence is uttered by Lolita, not Humbert Humbert.
[FIS-DS] No, abdominal.
Again, the same arguments apply as for the previous two sentences. This must be Humbert Humbert replying to Lolita's question.
[FIS] We would have to hang around for a while.
This is prototypical FIS. No reporting clause and a mixture of features appropriate for IS (3rd-person pronouns are appropriate for the narrating situation) and DS (the modal verb construction 'would have to' is appropriate for the character-character DS speech situation).
[FIS] The hospital was in the country, near the gay town of Lepingville, where a great poet had resided in the early nineteenth century and where we would take in all the shows.
Similar comments to the previous sentence apply here. At first sight, it may appear that the first part of the sentence is narration, but the last part is clearly FIS, and then the rest must also be consistent with that FIS interpretation to make sense.
[FIS] She thought it a peachy idea and wondered if we could make Lepingville before 9 p.m.
In grammatical terms, the subordination of reported clause to reporting clause, the third-person pronouns and past tense at first make this look like IS. But 'peachy idea' is clearly Lolita's own phrase, not the narrator's, and so this sentence is a mixture of IS and DS features and is really FIS.
[DS] 'We should be at Briceland by dinner time,' [N] I said, [DS] 'and tomorrow we'll visit Lepingville.
Note how the reporting clause is medial between the two DS stretches.
[DS] How was the hike?
[DS] Did you have a marvellous time at the camp?'
[DS] 'Uh-huh.'
[DS] 'Sorry to leave?'
[DS] 'Un-un.'
[DS] 'Talk, Lo, - don't grunt.
[DS] Tell me something.'