Zed71 at what time of winter will you be coming in?
The Finns usually don't start Ice Fishing themselves until it starts to warm up at the end of winter(it is just too brutally cold to sit out site for a long time, and the fishing gear freezes quite a bit in the colder days).
Looking through the info i could find it seems that most of the farms have stopped offering their own trips and focus on doing trips for the various Safari Operators.
Only safari operator that to my knowledge operate their own reindeer farm are Wild North but unfortunately they do not have a winter program on their website at this time. Also keep in mind that later part of December and all of January will be quite cold, and reindeer plod along(so no adrenaline filled rides there) so doing it all day may end up being too cold and just plain boring for the later part.
Look at something like 3-4 hour safaris as that is much more realistic(but keep in consideration what is included in the time, if that includes getting dressed(in Rovaniemi) and then also a drive to and from the reindeer farm that is going to make the actual reindeer ride shorter).
Wild North, Safartica and Lapland Safari's are the big 3 in town. Personally i would skip Lapland Safari's as they are the most "commercial". With either of the other two you can still expect to have to pay specially for "solo" trips if you do not want to be put into a larger group(does not always happen, but can).
Most of these companies will have something like a snowmobiles and ice fishing combination. But can always ask about Ice Fishing only if that is you main interest.
Ice Fishing is quite simple actually, it's just a small lure on a small ice fishing rod and a hole in the ice and then it's all about patience(and endurance if you go at it during the colder months).
You might want to look at getting the Ice Fishing and Reindeer trips from the same company, see if you can get some sort of "package deal" or discount.
One strong suggestion, stay away from Lapland Welcome. They do offer "all day reindeer safaris" but they actually operate so far from it that it's about and hour drive each way to and from and that is included in their timeframe, so is getting dressed. Also that company has a VERY bad reputation that it has earned and keeps earning(you can find plenty of info on that both on this site and on youtube).
Hope some of that helps!