You can kite off East Coast Beach in July and August. The winds will be from 12 to 18 knots. Also you can take a one hour ferry to Bintan Indonesia. Most will stay the night. The winds in Bintan may reach 20 knots and the water is nice and flat.
You can fly to Bali which is 3.5 hours away. The winds are similar to Bintan. But its a beautiful island and I highly recommend a visit.
Thailand is just 1.5 hours away by plane and a couple of hours bus ride to the beach. I don't kite there so I won"t comment. But Thailand is another beautiful country that you shouldn't miss.
The North East monsoon starts in late December in Singapore and Bintan but much earlier the further up North you go. So you may consider flying off the Vietnam as well. Another beautiful country.
Kiting in Singapore happens at two beaches, East Coat Beach which has no ban on kiting and Changi Beach which is currently banned but under review. Hopefully the ban will be lifted (but with restrictions this coming Jan / Feb 2014. There are other beaches where you can kite but they are not so popular.
Singaporeans also take a 2 hour drive into Malaysia for a day of kiting. Other than that most Singaporeans travel to kite. Budget airfares are affordable here and many wonderful countries to kite in are just 2 to four hours away.