You should eligible for your licence on the 22nd of December as this is 6 months before you licence expires. It's a simple mistake to make but January (1st month) is 5 months before June (6th month), not 6 months, i.e. 6-1=5, so that's 5 months.
I loved this trilogy. The ending was a bit unexpected... but I did enjoy the time spent speculating on what my daemon would be. I always liked the idea of a snake, usually harmless but potentially deadly.