I am going to attempt an answer from a linguistic angle. It is true that the Korean language and the Japanese language are regarded as two language isolates by many scholars. But efforts to bring them together have not stopped.
You may have heard the term Altaic languages. As one of the early and most hopeful attempts to bring Japanese and Korean together under the rubric of a large overarching language family called the Altaic family. The existence of this umbrella language family is not universally accepted, but since you apparently want something that shows the opposite side of the proclamation that "Japanese and Korean are language isolates", I am going to do just that with arguments that connect these two languages.
Such works as Roy Miller's seminal 1971 book Japanese and the Other Altaic Languages Chicago: University of Chicago Press and Nicholas Poppe's 1965 book Introduction to Altaic Linguistics argue that Japanese-Ryukyuan and Korean are the fourth and fifth branches of Altaic.
One paper I have found that presents a highly cogent argument for viewing Korean and Japanese in the same light is IGARASHI Yuko, 2003, Japanese as an Altaic language: An investigation of Japanese genetic affiliation through biological findings . Some passages from that paper:
In order to determine whether two or more languages are genetically related, the following aspects are generally examined: 1) whether languages have in common a large number of basic vocabulary items and morphological components through regular phonetic correspondences, and 2) whether there are many examples of identical semantics in lexical comparisons.
She argues Korean and Japanese share similar agglutination patterns. She gives three examples: Korean, modern Japanese, old Japanese, all romanized, so I won't attempt a reconstruction of the examples in my answer.
According to Poppe (1965), the agglutination of Altaic word inflection is characterized by adding suffixes mechanically to the stem, and each suffix has a single function. Both Korean and Japanese show this pattern.
She also argues Korean and Japanese have similar postpositions that are characteristic of Altaic languages. She then goes on to claim:
Martin (1991) describes the syntactic similarity between the two languages as follows:
The syntax is a model example of the object-verb language, with modifier preceding modified, with the predicate at the end, and with the relationship between the adjuncts (the noun phrases) and the predicate shown by postpositional particles, by ellipted postpositions, or (as with adverbs) left unmarked (p. 281).
...
The examples above demonstrate that it is possible to translate word-to-word and morpheme-to-morpheme between Japanese and Korean. Therefore, Japanese syntax is remarkably similar to that of Korean.
