I will be at Harbin in January 2016 and plan to go to Jilin for 3 nights.

Yes, you can take bullet train from Harbin West Railway Station, and arrive at JL directly. The trains depart at 07:18, 10:10, 13:24 and 15:56.
The duration is about 2 hours. The ticket fare of First class seat is CNY215 and second class seat is CNY141.

You can buy the tickets when you arriving at the railway station. But it's suggested to book in advance in case the tickets may be sold out. TravelChinaGuide provides ticket booking service. If interested, the contact information is bookingtravelchinaguide.com.