Bismarck is very famous for his quotes, but even the Otto-von-Bismarck-Stiftung has no complete summary of them (at least online).

Asking them would be your best option for a reliable result, since his quotes are often used out of the original context.

_{Note that this answer was written before "Update 2" in the question appeared.}

Summary

It seems to be an anecdote, reported only one single time, as "he once said", by one source, undated. It is of a singular nature, not repeating. In that single source it has no relation to any specific occasion and is even phrased with ambiguity, possibly even with an intended humorous effect.

It seems to be reported first by Tiedemann in a speech in 1897 and after Bismarck's death this seems to have spiralled out of proportion. That some admired or despised Bismarck as 'a hater' made for a well fitting 'quote' in both cases. But Bismarcks own words – even if correctly recorded by Tiedemann – make for very slim evidence to diagnose his character as 'frequently insomniac and busy hating through the night'. But the single source Tiedemann is not of unlimited value anyway, so this quote has to be read with a certain pinch of salt at the ready.

Since the beginning of the 20th century this 'quote' makes the rounds.

This was found attributed to him,

• variously 'writing to his wife',
• into his own diary,
• "once said to Tiedemann", and
• doubted as unproven anecdote.

Could we prove at least some of these at the real source? For example in his letters and/or Tiedemann's memoirs which are for a large part online at archive.org (example). Although mostly written in fraktur and catastrophically OCRed.

That he might have said this in various forms ("gehaßt"/"durchgehaßt)", might sill be true. But another possible explanation for that could be that he simply said it that often, varying it himself. And that then being recorded differently and even accurately in different sources. But that seems not to be the case.

If it's in his diaries in one form, in a letter to his wife in another, in memoirs of Tiedemann in yet a third, then we would have three primary sources, two of top quality.

One such source is indeed

That scanned image shows what Prof. M Lappenküper's, from the Otto-von-Bismarck-Stiftung, believes is the main source. It reads:

Bei aller Kraft und Erregbarkeit seines Temperaments blieb doch ein kühler Realismus der Grundzug seines Wesens. Er sah die Dinge wie sie wirklich sind, unbeeinflußt durch schwächliche Empfindsamkeit. Er mag die ganze Skala der Gemüthsbewegungen durchgemacht haben und zwar mit der vollen Wucht seiner Persönlichkeit ("ich habe nicht schlafen können, ich habe die ganze Nacht gehaßt", sagte er mir eines Morgens) – ich glaube aber nicht daß er jemals sentimental oder pathetisch geworden ist. Wie jede Phrase war ihm jede Pose verhaßt und Posiren würde er es genannt haben, hätte er eine weichliche oder künstlich forcirte Stimmung zur Schau getragen.

Jetzt liegen alle jene Erinnerungen in nebelgrauer Ferne hinter mir. Die Bilder, die sich einst farbenprächtig dem Gedächtnisse eingeprägt, beginnen allmählich zu verblassen[…]_{(p52)
–– Christoph von Tiedemann: "Persönliche Erinnerungen an den Fürsten Bismarck. Vortrag gehalten in der historischen Gesellschaft für den Netze-Distrikt in Bromberg am 18. November 1897", S Hirzel: Leipzig. 1898. (p42)}

_{Translation: Despite all the strength and excitability of his temperament, a cool realism remained the main feature of his nature. He saw things as they really are, uninfluenced by weak sensitivity. He may have gone through the whole range of emotions with the full force of his personality ("I couldn't sleep, I hated the whole night," he told me one morning) – but I don't think he ever became sentimental or pathetic. Like every phrase, he hated every pose and would have called it posing if he had displayed a soft or artificially forced mood.}

That is a Tiedemann source, which Lappenküper thinks is reliable (seriös). But in this same pamphlet Tiedemann is freely admitting that it is a vague and fading memory from long ago. He kept a diary, but he doesn't get into specifics regarding the date or opportunity for this quote. Just "he once said it".
For comparison: In this same source Tiedemann goes into some vivd details regarding certain dates and affairs. The context of the 'quote' also makes it clear that Tiedemann is actively knitting towards a legend of his liking. Including hyperbole in anecdotes.

A note on the translation:

Ich habe nicht schlafen können, ich habe die ganze Nacht gehaßt.
I couldn't sleep, I hated all the night.

This is ambiguous in the original. It could as well mean that he could not sleep and (that because he) was busy hating during the night. But equally that he couldn't sleep that night and that (therefore) he hated the (sleepless) entire night itself. Meaning it was just a terrible night. Without more context, this seems undecidedable from the source. Even if the 'hating' reading should be the better fitting one, it cannot be excluded that this was meant as a joke.

More specifically, Tiedemann's own and much more detailed memoirs ("Aus sieben Jahrzehnten: Erinnerungen - Bd. 2: Sechs Jahre Chef der Reichskanzlei unter dem Fürsten Bismarck", S Hirzel: Leipzig, 1909) have 487 pages to pinpoint this sentence, yet fail to do that?

I didn't check Bismarck diaries. In Bismarck's memoirs he uses 'gehaßt' exactly one time, not related to his nights, but describing other people's attitudes.

As far as his letters to his wife are completely published in

• Bismarckbriefe. 1844–1870. Originalbriefe Bismarcks an seine Gemahlin, seine Schwester und Andere. Bielefeld 1876.
• Bismarckbriefe 1836–1872. 7. Auflage Bielefeld [u.a.] : Velhagen & Klasing, 1898
• Fürst Bismarcks Briefe an seine Braut und Gattin. Hrsg. vom Fürsten Herbert von Bismarck. Cotta, Stuttgart 1900.

A cursory reading does not reveal this quote to me, but quite some instances of him not complaining about sleeplessness. Instead we find examples describing how he sleeps "long" (Potsdam 14. Nov 1848) or "11 hours…" (when on holidays) or "not long but well" (Petersburg 3. May 1873). That he always (or at least frequently) 'hated through the night', and wrote that at least once to his wife, seems not true. As ups and downs come and go, he reports of occasional difficulties sleeping and also ample sleep over the years.

But if Tiedemann in this version just cited is the only source for this 'quote' than it can be called justly: 'maybe a true report of a personal witness' – and a legendary anecdote without further merit regarding Bismarck himself as well at the same time.

As a contemporary recension of this work remarks without all irony:

But how much more intimate is the connection in which Tiedemann stood with Bismarck, and how much finer and nobler he knows how to reproduce the changing images of Bismarck's day's work, how he complains and moans after sleepless nights, eats and drinks enormously, chats ravishingly, drives up like a lion, and then can probably be unapproachable and dangerous, and yet again and again into his 'real ruler's' mood of the balanced tension when willpower returns. Thus a truly heroic impression, because the effect of the good heroic epic is also based on the fact that we have a hero not only in his main and state actions, but also in all his humanities.
_{–– Historische Zeitschrift, 3 / 7 / 103, 1909 (my translation and emphasis)}

One other source that might be reporting this independently from Tiedeman might be Ernst Schweninger: "Blätter aus meiner Erinnerung" in: Erich Marcks & Karl Alexander von Müller (Eds): "Erinnerungen an Bismarck : Aufzeichnungen von Mitarbeitern u. Freunden des Fürsten, mit e. Anh. von Dokumenten u. Briefen ; In Verbindung mit A. v. Brauer / gesammelt von Erich Marcks; Karl Alexander von Müller", Deutsche Verlags Anstalt: Stuttgart, Berlin, 1915. (p 177–222, esp 216, 218). But this late source is noticeably different from what Schweninger wrote himself in: Ernst Schweninger: "Dem Andenken Bismarcks : Zum 1. April 1899", S Hirzel: Leipzig, 1899. In the early pamphlet Schweniger makes no mention of the sought after quote. That would make it all the more likely that he or the editors updated that memory, possibly again based on Tiedemann.

Answering the title question "Where did Otto von Bismarck say “laying awake all night, hating'?" one can only say with certainty: in Tiedemann's memory. How reliable that witness might be in that regard remains debatable. How much this one source contaminated the recollections of other sources coming after him is uncertain. We do not get to know from Tiedemann on what occasion this might have happened.