TLDR; Chiang privately admitted defeat before the civil war even ended, and everyone knew the KMT would not retake the mainland.

Actually, as Jay Taylor mentions in his book The Generalissimo, Chiang accepted quite early that he would never return to the mainland. As early as 1947 Chiang and his wife Mayling toured the island under the guise of a vacation, but most likely as a way of scouting out suitability for moving the Kuomintang there. That same year, Chiang appointed a close ally as governor of Taiwan, and the newly founded CIA set to task helping Chiang estimate whether a KMT takeover of Taiwan would lead to rebellion. They eventually ruled that, no, it wouldn't, and this would end up one of the CIA's first intelligence errors.

The propaganda surrounding the KMT's triumphant return to the mainland was actually somewhat more powerful inside the USA rather than Taiwan. When Chiang moved to the island, there was always that small shining light that one day the KMT would retake the mainland, but he, his officers, and his increasingly important US advisers knew this was impossible. Instead, Chiang and the KMT focused their efforts on retaining legitimacy as the official representatives of the Chinese people to the UN, something awarded to him after WW2 and that Taiwan held until 1971.

In America, Taiwan, like South Korea and then eventually Vietnam, became a useful political piece in the game of Cold War chess that was unraveling in East/South East Asia as communism spread across the continent. One of Eisenhower's minor talking points when he ran for president was that he would "unleash the beast (Chiang) into the underbelly of communist China," arguing that Truman had held Chiang back from doing so. There was a small, but loud group of Republican senators rallying around Chiang as a symbol of anti-communism in congress, but not enough to actually garner any real support other than loans and materiel aid here and there.

The KMT and CCP faced off in a few skirmishes along the minor islands between Taiwan and mainland China, mainly Quemoy and Mazu (Matzu). But on both sides the skirmishes were just saving face, and all entities involved knew that. Mainland China had two issues; first, the KMT took most of the navy with them, and since the CCP was based out of Manchuria and Yan'an before taking over the rest of China, they were landlocked and had no opportunity to develop a navy other than a few junk boats. Second, the mainland was severely underdeveloped from decades of war.

Things didn't go so well in Taiwan either. In May 1949, just as the KMT was settling into Taiwan and importing massive amounts of mainlander Chinese with it, riots broke out in Taipei as the Taiwanese expressed their displeasure with being the chosen one for the KMT. Chiang reacted quite swiftly; he lost power once, he wasn't going to be exiled from any land he considered Chinese again. KMT troops killed thousands of protesters, and martial law was declared. The martial law, sometimes called the White Terror, lasted 38 years until Chiang's death. During most of this period Taiwan's political situation became a lot like South Korea. Chiang was despotic, restricted free media, and favored mainlanders over the Taiwanese. And quite possibly more importantly, the KMT was still reeling from a hyper-inflated currency it brought with it from the mainland. Economic progress in Taiwan, though, did manage to expand rapidly, thanks to the efforts of the KMT to industrialize the mainly agrarian island. Nowadays Taiwan is a regional economic power, but this rise was gradual and didn't really come about until the mid-80s, much like South Korea.

1950 was a pivotal year for both the CCP and KMT. For Mao, he had to make a decision. Either invade Taiwan while America seemed at its weakest to intervene, which seemed like an OK option as North Korea pushed the allied forces into Pusan, or wait it out and see what happens in Korea; ultimately as we know Mao felt forced to intervene on NK's behalf, effectively delaying any sort of invasion for the near (and as we know far) future. For Chiang, 1950 was as much an opportunity as it was a risk. On the one hand America was no longer interested in protecting Taiwan as long as Korea was going on. On the other hand, Chiang could send troops to Korea to prove his loyalty to America and the UN, but it would weaken his already weak position in Taipei. Chiang sent troops to Korea in any event, and so did Mao, leaving Taiwan in a semi-secure position.

Over time as the Cold War heated up, Chiang rested comfortably in Taiwan, especially after the US declared they would patrol the strait between the mainland and the island. By about 1955 or so, and definitely by 1960, it did become increasingly clear that Taiwan would never retake the mainland unless something miraculous occurred. Additionally, the US made it clear that they didn't want Taiwan meddling in mainland affairs, the CIA produced Taiwanese agents for the purpose of counter-intelligence on the island, and spying along the southern coast, but nothing more. As US involvement in Vietnam also expanded, Taiwan again took a backseat in US interest. Much like Korea, Chiang again sent troops to Vietnam for good faith, and America continued to supply the KMT with state of the art aircraft, anti-air, and other materiel to secure the KMT position over the island, but nothing more than that.

In more contemporary times, the KMT's claim to legitimacy over the mainland has become increasingly irrelevant and weak. In fact after Chiang's death and the liberalization of the island under his son, the KMT has quickly lost a lot of its support. While Taiwan continues to be a close East Asian ally of America and the West more broadly, it has more zealously asserted its independence as a nation, which in the long run means denouncing any claims to foreign governance. For the foreseeable future, the KMT will remain on Taiwan.