The first Nina is 101010, which is 42 in binary (in “bits”—i.e., “binary digits”).

The second Nina is FORTY-TWO, which is 42, “fully spelled out.”

The connection of the Ninas to the title/prelude is that Rule 42, as proclaimed by the king of hearts in Alice’s Adventures in Wonderland, said: “All persons more than a mile high to leave the court.” This arbitrary rule was intended to be used by the king to evict Alice, who had been rapidly growing in height (hence “it aimed to exclude a rapidly growing class of population”). The puzzle prelude refers to “this king’s rule” rather than “this king’s rule.”

Alice resisted the rule and did not leave the court. ALICE is indeed the “successful resister” located “roughly” at squares 3, 7, 83/7, 13, 23. Here 83/7 is 11⁶⁄₇, identifying the 6th of the 7 white squares spanning positions beyond 11 upto 12. The letters at these five locations, respectively, are ICLAE. Anagramming them, as indicated by “roughly,” yields ALICE. I expect many solvers to figure out ALICE first and then work their way backwards to see what “King’s Rule” refers to.

The number 42, of course, is also the answer to the ultimate question of life, the universe, and everything, in The Hitchhiker’s Guide to the Galaxy and its sequel novels. Most clue surfaces in this puzzle (all but 8a, 11a, 14a, 27a, 3d, 16d, 17d, 19d, 24d) refer to these books. “DNA”—used in a couple of clues—are the initials of Douglas Noel Adams, the late author of these gems.

Additional Notes (contain spoilers for the HHGTTG books)

• 13a. Zillionaire Zaphod retreated embracing a Caucasian person (5). Zaphod Beeblebrox left a party with Tricia McMillan, aka Trillian, who then left earth with Zaphod before earth got destroyed by the Vogons.
• 18a. Gogrilla M., entrepreneur bearing regret (6). Gogrilla Mincefriend was an entrepreneur who re-invented the staircase when prescient elevators refused to move as they were often terrified of what lay in store.
• 30a. Draw world like Lamuella, suited well to idyllic, laid-back terminations (8). Lamuella is an idyllic planet where Arthur Dent spends time as an excellent Sandwich Maker. The planet seems like a nice place to spend one’s final days.
• 1d. Charge with abduction I, Zem, unmasked (6). Zem (a sentient mattress) is the only witness to the abduction of Marvin by the Krikkit war robots.
• 4d. True, Stavro Mueller finishes ailing all over the place (8). The event at Stavro Mueller Beta, a nightclub in London, ends the lives of all remaining survivors of earth’s destruction by the Vogons.
• 5d. Representative rewired Colin after removing the last bit of towel, getting InfiniDim credit openings (6). Colin was a flying security robot, trapped using a towel by Ford Prefect. Ford rewired Colin to get free InfiniDim credit card payments. This was a long clue and the surface only makes sense if one knows this obscure part from the books. I had proposed a slightly simplified rewrite that did not make it: Representative rewired Colin after removing the last bit of towel, I see (6).
• 10d. Hotblack Desiato demise is to block a redistribution (5). Hotblack Desiato, a rich rockstar, indeed spent a year dead for tax reasons.
• 16d. The extent to which rock can absorb pop? Limitless! So, I try, desperately (8). This was another long clue with a clunky surface. I had an alternate clue lined up, that also did not make it: The difference between hard and soft rock? (8)
• 23d. DNA is back after adding top record-breaking "Restaurant at the End of the Universe" to trip! (6). I was happy to get the entire title of one of the HHGTTG books into the clue.
• 26d. Sperm whale starts yearning for change of direction (6). The Magrathean sperm whale, created mid-air by the Infinite Improbability Drive, had a short life span of about a minute before it hit the ground and died. It probably did yearn for a change of direction.