I fear that there is no such single work that attempts to, in the course of exposition, touch all the aspects of English grammar.
To attempt to interpret your wants as loosely as possible, you'd like any literary work that uses (not mentions) grammatical aspects, somewhat in analogy to the Chaos poem which exposes a number (but not all) of idiosyncrasies of English spelling.
This necessarily would attempt some kind of constrained writing, in the manner of Oulipo, which is taking some arbitrary composition rule, and writing an entire work that doesn't break the rule.
There are many examples of such constrained writing, but none that I know of that attempt to run through all grammar rules of English intentionally. Generally those works tend towards either spelling, like 'A Void' which doesn't use the letter 'e' and 'Ella Minnow Pea', an epistolary novel which drops letters used one by one in each successive chapter (and this is meaningful to the plot), or to style, like 'Exercices de Style'](https://en.wikipedia.org/wiki/Exercises_in_Style).
These rules seem to be nowhere near grammar. But the lack of an 'e' does eliminate all use of '-ed' past tense words, which is a huge grammatical restriction.
Then there are works which are not exactly part of the mostly francophile Oulipo movement but still address grammatical things, though somewhat narrowly. Most literary works are past tense and reported (talking about other characters). But a few works use just the present tense (or some sci-fi works which play with neologistic tenses for speaking about time-travel. And other works play with the second person (all talking to the reader about the reader), or I suppose some works could be considered written in first person (naturally somewhat autobiographical or stream of consciousness).
Not at all what you were asking but possibly may end up being relevant are works that use a lot of eyedialect, that is written in a way that if pronounced out loud gives the accent of a local dialect. Mark Twain's 'Huckleberry Finn' is famous for writing major characters with distinct local dialects. Then there is Irving Walsh's Trainspotting, where all dialog is in Scots or Scots English (narration in standard English).
I realize this is not the direction you say you want, but I'm telling you that it is most likely what will get what you want, except not in a single written work.
But...
Even though they are didactic, explicitly intended to teach grammar, the set of Schoolhouse Rock songs that cover grammar and parts of speech with accompanying short videos:
Schoolhouse Rock!
may well be the closest to what you want. It only covers the subjects minimally in the very outdated/inappropriate latinate seven parts of speech. These are all very entertaining but still at a very elementary level of grammar. For example, it might be nice to have one on phrasal verbs (verbs that come with a preposition which changes the meaning greatly, e.g. call off/call out/call in, put up/put out/put down/put on, etc), or how modals differ (can, will, may, might, would should, etc).
