Welcome. Don't miss North Sikkim at any condition. Book a 3D/2N tour from local tour operators at Mall road. Believe me, It is an untouched paradise on earth. On 3rd day you will be back to Gangtok. Then you can go to Pelling.
I clearly temember that when i went to Lachung in the month of May i expirienced snowfall and that lake was frozen . As of high atlitude of 12000 ft . it becomes difficult to breathe due to low oxygen levels . Hence, it is highly recommendable not to visit the place in the month of January as even the locals come down to lower atitudes during these days. Yet if you want to expirience cold then you should pay a visit in the month of March. Happy Travelling !