SRQ:
You will have very limited daylight (4-5 hours/day) that time of year. Almost everything is closed on the 25th (and much on the 26th). Tours do operate on all days of the year.
As you mention, driving can be very dangerous in Arctic winter. I drive in New England winters all the time, but I learned a lesson about Mother Nature in my one winter (March) driving experience in Iceland.
Northern Lights can be incredible, or can be wispy grey streaks (I haven't seen 'em), but do require hours of standing outside in the winter night. Your 4 year old won't enjoy this. Your 11 year old will probably not enjoy this.
I'd consider a multiday tour if you are set on this, but one where you can leave one/both kids somewhere for northern light hunting. A 3 day tour of the South Coast and/or a 2 day tour of Snaefellsness should work for you