Greetings! Unfortunately you have hit on a rather obscure question here, and one whose historiography is either mostly in German, Dutch, far too old to find online, or a combination of the three. Let's see what light I can shed however, based on the limited English and somewhat more up-to-date publications on the matter.
The first reason is pretty simple: the Netherlands was the country chosen for Wilhelm II for lack of a better choice. Switzerland in the south was deemed too international and too diplomatic in its political atmosphere. Not to mention the small fact that getting the Kaiser there would mean either crossing enemy lines in war (not something you want to do with a head of state), or traversing German districts which were heavily influenced by left-wing rebels (or so at least, it was reported). Belgium was out of the question, because it was still possible that Allied troops would push further and threaten the Kaiser with capture (or worse) before the war ended. Denmark and the Scandinavian countries would necessitate long voyages (and in the case of Norway, a dangerous naval trip through the British blockade, which would almost certainly catch the Kaiser's entourage). Holland at least, still had a monarch in the figure of Queen Wilhelmina, who was also (helpfully) a distant relation of Wilhelm II by way of the House of Orange (the Dutch royal family tree which had German relatives in the house of Hohenzollern). From the German army headquarters in Spa (Belgium), to the Dutch border, a distance of about thirty to forty miles was all that the Kaiser had to negotiate, much if not all of it controlled by disciplined German troops already.
As for why the Netherlands allowed Wilhelm to stay in-exile there, the reasons are a mix of royal ties between Queen Wilhelmina and Wilhelm II, as well as "the Dutch tradition of hospitality and asylum", though the civilians did not share this opinion of the abdicated Kaiser. When the royal procession stopped at Eijsden station on the way to exile, the Kaiser endured about an hour's worth of shouts from Belgian refugees and locals. Common phrases heard (as recored by eyewitnesses) were: (and I apologise for such foul language, but all in the name of historical narrative) schweinhund (bastard), mörder (killer), and the somewhat satirical Kaiser, wohin? Nach Paris? (Kaiser where? To Paris? - likely a jab at Wilhelm's dream of marching through Paris in 1914). Upon arriving at Maarn station in the province of Utrecht, the Kaiser was received by Godard Bentick, Graaf van Aldenburg-Bentick (member of a distinguished Anglo-German-Dutch family), who housed him at the estate in Amerongen for three days.
Such was the atmosphere and rationale (as far as my source-work and research, limited by lack of current focus on the matter in academic circles) for the Kaiser's choice of exile. Hope this somewhat helps, and sorry for the lack of further detail!
Sources
Howard, Michael. The First World War: A Very Short Introduction. Oxford University Press, 2007.
Marks, Sally. ""My Name Is Ozymandias": The Kaiser in Exile." Central European History 16, no. 2 (1983): 122-70. Accessed January 31, 2021. http://www.jstor.org/stable/4545982.
An extract from Count Detlef von Moltke on the abdication, originally quoted in Source Records of the Great War, Vol. VI, ed. Charles F. Horne*, National Alumni 1923.* Found here (free access)