There are a few different ways to go.
1. Fly Singapore airlines (via silkair) directly to Langkawi. As you seem to have already checked the only thing they have available on their web site is on the 13th and 17th. Price on the 13th was SGD 266 just for reference.
2. Fly Malaysia airlines to Langkawi from Singapore. Just checked price and it was SGD 275 and they have flights on the 15th
3. Take taxi or bus to Johor Baru and fly air asia from JB to KL then KL to Langkawi. Car to JB is about SGD 50 bus is SGD 10, flight from JB to KL is RM 9.99, and flight from KL to Langkawi is RM 79. They also have vacancies on the 15th.
4. Alternatively you can fly into Penang and take the ferry from Penang to Langkawi. Its not very long, under 2 hours and costs RM 40 per person. It can be a fun experience also. Singapore airlines flys to penang directly (Via MAS). Cost is SGD 231 via singapore airlines web site and SGD 240 via MAS web site.