Nemurenai aru is an example of a Chinese-ish Japanese fobism particular (as far as I can tell) to characters that frequent a certain storyscape in Japan, namely, anime and manga. Nemurenai means “can’t sleep” and the fobified element is the unnecessary “aru” tacked on at the end, a kind of misused copula (like desu in normal Japanese, dearu in written Japanese <– also a viable culprit for the fobism, ye 也 in classical Chinese, all of which most often get approximated as the verb to be in English).
In my relatively limited years spent engaging this 2D world, I have noticed that it comes up mostly in comic/gag series, such as the case of Kagura in Gintama (from which this tab title is directly lifted) and China in Axis Powers: Hetalia, but I’ve heard it also goes back to older series. In these cases, the peculiar sentence ending functions as a part of character design that vaguely marks Chineseness (Kagura is coded Chinese in various ways, and metonymically referred to as China by Okita, but is not actually from a place in the series referred to as or resembling China). Although friends who have resided in Tokyo all admit to understanding this as part of a stereotypical “Chinese” accent in popular representation, no one apparently has ever met or heard of a native Chinese speaker speaking Japanese with this sentence ending.
Messing with sentence endings or designating speech patterns is a pretty crucial way of building character in anime/manga, but it tends to get out of hand in crack series like Gintama and Hetalia, where basically anything goes. Mayo (short for mayonnaise) becomes a sentence ending when one Hijikata from Gintama poses as Prince Mayo from the planet of mayonnaise to get out of a bad romance (episode 114). A more common example might be nyan (meow), which is often used by cat-like characters: characters sporting cat ears (nekomimi) and/or tail or characters otherwise trying to invoke the particular moe induced by this type of cosplay (see Kikumaru from Prince of Tennis).
Parody speech: “Today, Mimi-chan (<–referring to self by pet name in 3rd person) went to the convenience store-nyan! We (<– I and my third person self) were looking forward to the new limited edition Pocky, but they were sold out…Mimi-chan is super disappointed-nyan!”
Ok, so we get that this is about character design and excess.
What is the exact effect of “aru, aru”? Well, I’m no expert on this and if anyone out there has more examples of characters who speak like this or theories about how this all got started, please share!!! BUT, if I were to venture a reading based on my gut feeling, I would say that, like a copula, it can make whatever is being said sound like the speaker is stating something in a very definitive, well, yes, overstated, way. I’ve read Gintama fic where writers will add a “yup” or “for sure” or some sort of short affirmative expression at the end of the sentences in her spoken lines. I think it captures the feeling of “aru” quite well-aru.