Yes, sleep when you need it.
Wake up later in the day and take the opportunity to explore some of Singapore's night activities. Start with the night safari at around 6-7pm, then go and see the Christmas lights in Orchard Road (they will be up in December), then have dinner later at one of the hawker centres, or a late night restaurant. Singapore is one of the world's 24 hour cities, so there shouldn't be any problem finding something to do later on, even with a 10 yr old in tow.