I have bad news for you. If you want to make good shots in those conditions, you are going to have spend a lot of money.
A lot. You need a 70-200 f/2.8. The
previous generation Nikkor, which happens to be the one I use, goes for $2,100. My apologies. On the bright side, that is about $300 less than what I paid for mine over 6 years ago. Amazing. Same great lens, six years later, $300 less. The
new version of the 70-200 goes for $2,800. Your pictures will not be $700 better, so I would get the older one. In another plus, if you put the 70-200 on your D5500, it has an effective reach of 300 mm due to the crop sensor. The 70-200 is an absolutely amazing lens and even after owning it for 6 years, I still marvel at how awesome it is. Unless photography is really your jam, however, it is probably too pricey.
In the realm of what semi-reasonable people might consider, there is always the
85 mm f/1.8. It is faster than the 70-200, but it doesn't zoom, and does not have the same reach. It costs $475. It will get you a little further down the ice than the kit lens, blur the background, work well in low light, and will look great.
As I said, there is no one lens to rule them all. Pick your poison.