A good choice is a Mediterranean climate.
A Mediterranean climate or dry summer climate is characterized by dry
summers and mild, wet winters. The climate receives its name from the
Mediterranean Basin, where this climate type is most common.
Mediterranean climate zones are typically located along the western
sides of continents, between roughly 30 and 45 degrees north and south
of the equator. The main cause of Mediterranean, or dry summer
climate, is the subtropical ridge which extends northwards during the
summer and migrates south during the winter due to increasing
north-south temperature differences. (ref)
The wet winters and dry summers aren't necessary to your situation, but they come part and parcel with mild summers and winters and basically gorgeous year-round weather. Avoid the more extreme examples of the Mediterranean climate and aim for, say, coastal California.
In particular, try Catalina Island, not far from Los Angeles.
Santa Catalina Island has a very mild warm-summer Mediterranean
climate (Köppen Csb) with warm temperatures year-round...The average January temperatures are a maximum of
58.4 °F (14.7 °C) and a minimum of 47.6 °F (8.7 °C). Average July temperatures are a maximum of 78.1 °F (25.6 °C) and a minimum of 60.0
°F (15.6 °C). There are an average of 12.5 days with highs of 90 °F
(32 °C) or higher and an average of 0.3 days with lows of 32 °F (0 °C)
or lower. (ref)
This particular island is fairly small ("22 mi (35 km) long and 8 mi (13 km) across at its greatest width") so yours would need to be bigger in order to accommodate all but the smallest of nation states.
Mediterranean climates occur on the Western coasts of large land masses in the correct latitudes. Putting your island further east will completely change the climate. Between Korea and Polynesia is the Philippines (well, mostly the Philippine sea to the east, which has room for a good sized island invention).
The Climate of the Philippines is either tropical rainforest, tropical
savanna or tropical monsoon, or humid subtropical (in higher-altitude
areas) characterized by relatively high temperature, oppressive
humidity and plenty of rainfall. (ref)
Even if you go much further north, to a attitude closer to Southern California, you still have more summer heat.
The climate of Fukuoka, a Japanese city located on the north coast of
Kyushu (which is the southernmost of the major Japanese islands), is
temperate humid, with quite mild winters and hot, moist, and rainy
summers. Like the rest of Japan, the city is affected by the monsoon
circulation: in winter, the northwest cold currents prevail, while in
summer, they are replaced by hot and humid currents of tropical
It all depends what you want. If location is more important to you, then a humid temperate or tropical climate should still be within the range of more moderate winters (if not summers). But if you're okay with going further east, then nestle your island along the southern Californian coast and enjoy very mild, near perfect, weather year round.