Ya gotta distinguish two kinds of dependent clauses (at least -- English demands three, at least). Here the relevant issue is about truth-value or not. In 'mi lukin lape e ni: mi tawa mun kepeken luka mi', the second sentence (they are separate sentences in tp, not parts of one sentence, but...) is not strictly asserted and so it is not considered whether it is true or not (or, at least, the fact that it is false is not counted as falsifying the rest of the passage). On the other hand, in 'mi pali e ijo ni tan ni: mama mi li wile e ona,' the second part is checked for truth and, if lacking it, tends to falsify the context, at least partly (I still made it but my reason doesn't apply). tp makes not distinction between the two, except that the first sort are almost always ultimately the objects of verbs of cognition or communication (toki, pilin, lukin, kute, sona, ...) and the others come in in other ways. (A word of warning here: the sentences before 'la' are not asserted either but rather serve as triggers for when the following sentence is asserted and the whole is true or false depending upon what the after 'la' part does when the trigger is pulled.) The first sort of "subordinate sentences" also come in two versions, direct quotes and indirect quotes, exactly what was said (or a trat of it) and paraphrase (with pronoun shifts and the like -- tense shifts where available). Only the first of these deserves quotation marks (and the intro 'nimi ni' if you are fussy).
Stringing out 'ni:'s is not only bad form, it is hopelessly confusing and calls for a bit of rethinking, different for each occasion, of course.
Sentences of the second type also come in a variety of special cases. There are, first of all, the various causal cases ('tan ni' at the beginning or end of a sentence, for example); these need no further comment. Secondly there are the identifying and additive clauses: "the man that came to dinner" and "the man, who came to dinner" (last year's rule book, I don't know what the rule is today), restrictive and non-restrictive relative clauses in English. If the restrictive sentence is false, the identification may fail (or may be fine since the hearer shares your false information or just ignores the claim) but the rest may be OK (with the intended referent, whether properly salvaged or not). In the non-restrictive case, just the false claim about the person is false, the rest goes on as usual. We can add complications upon complexities here, but this is more than enough. Stylistically, the usual suggestion is that restrictive relative clauses precede their referent and are picked up in the main sentence by 'x ni' where x is the salient noun in the clause (separate sentence). Non-restrictives tend to follow, with 'x ni' in the following sentence, picking up the salient noun in the main.
The big problems are less strings of 'ni' than simultaneous 'ni's, the "He told me this because of that" sort. The best rewriting is, as always to get each thing into a separate sentence. "He told me this. He did that because of this" Of course, each case requires its own variation on this theme. And, yes, it would be nice to have some punctuation to show the scope of these various moves, but none practical come to mind (leaving out 2-D writing for the moment).