Without natural cycles to base periods of time on, humans would simply use multiples, and/or man-made events. Humans love order and categorization, and in fact our survival depends on it; I suspect this would likely be the case for any sentient civilized species.
Weeks primarily exist because of celestial observations:
The Babylonians, who lived in modern-day Iraq, were astute observers and interpreters of the heavens, and it is largely thanks to them that our weeks are seven days long. The reason they adopted the number seven was that they observed seven celestial bodies — the Sun, the Moon, Mercury, Venus, Mars, Jupiter and Saturn.
Simply use multiples
Without these, humans would probably lump days into groups of five or ten (because 5 fingers on a hand or 10 fingers on both make base-5 and base-10 relatively intuitive systems for us). Similarly, "years" might be groups of 10 weeks (so you'd have 100 days to the "year"). Beyond this you'd have decades, centuries, and millennia. If you had a different sentient species that had a different number of digits, their base system would likely somehow be based on that.
Man-made demarcations of time
Alternatively, if the government of your island is a single nation under a hereditary governmental system, perhaps each ruler's reign, and for larger epochs, a given dynasty: so "in the 15th year of Hamaz Magnus" would be understood to indicate the period of time ~1500-1600 days after the coronation of Hamaz Magnus in the Magnus Dynasty. Dynasties tend to rise and fall (revolutions, coups, lack of familial heirs, etc) so they will be irregular – you might have some where a given dynasty only has a single regent: if Hamaz Magnus was appointed the next heir by the childless Damek Brevel, but was in turn overthrown by his military advisor Jeupor Muron, the "Magnus Dynasty" would still be understood to be between the Brevel and Muron dynasties.
Of course, when speaking or writing more casually, no one will say "on the third day of the tenth week in the fifteenth year of Hamaz Magnus"; there will probably be some (at least quasi-)standard abbreviated notation, like "MagHa-15-10-3" or perhaps just "MagHa-15-33" – with only 100 days to the year, a year-day notation isn't horribly complex. To anyone used to this system, it's clearly Dynastic Name, Ruler Name, year, and day. Note that "MH" might be too ambiguous, especially if Jeupor's granddaughter were named Hicias in this example, so they'd probably include a few letters to disambiguate. Of course people being people, poor Hamaz might lose his surname and just get lumped in at the beginning of the Muron dynasty 😂