Anyone who wants to see the northern lights needs to understand a little bit about them, and it seems so many people just think all they need is a ticket to Iceland. Alas, not true. To see the northern lights - from anywhere - all of the following need to happen:
- It needs to be dark outside
- The sky needs to be clear
- There needs to be a period of solar activity - this one is complicated
- It certainly helps - but is not a requirement - to be someplace way up in the northern latitudes ("up north"). Nome is as good as Reykjavík.
Lets look at each of these...
It needs to be dark outside. Duh. This is critical. That means get away from cities and other places with lots of lights. Remote Anyplace is where you want to be. Winter is better, but not because of some magic that produces more aurora in the winter: it's because the sky gets darker earlier and stays darker longer in the winter - that's all.
The sky needs to be clear. This is also critical. This is the inconvenient truth that has spoiled many a short trip to Iceland (and a million other places) to catch the aurora. Is it cloudy tonight? If so, sorry, you ain't gonna see anything in the sky expect perhaps snow falling. Try another day (night). Remember - in many northern places, winter brings with it clouds, sometimes for many days without a break. Iceland is no exception. If you are taking a trip to Iceland to see the northern lights, you would be advised to allow many days there to increase your chances of getting a clear night. Or else have other good reasons to go.
There needs to be a period of solar activity. This is critical, too. The northern lights are caused by "solar wind" (charged particles that stream out from the sun) which enter the earth's atmosphere near the poles (the same thing happens near the south pole, too yes, the "southern lights" aka the aurora Australis). The sun has long cycles within which is puts out less and then more solar wind. Broadly speaking, it's possible to predict these long cycles somewhat (like predicting the hurricanes for next year) but it's impossible to predict exactly how much activity there will be for any short period - like your 2 nights in Iceland. In other words, it's a complete crap-shoot if you're trying to guess when the aurora will show up. Nobody can predict that.
It helps to be someplace up north. This is certainly true. But if there's an exceptional aurora, it can be seen lots of places (and if there's no activity, you can be at the north pole on a clear night and see nothing in the sky but stars). I've seen the northern lights from my backyard in Seattle (faintly), in Alaska many times (pretty good). But the most spectacular I've seen was from near Detroit, Michigan, in a brightly lit urban area - because THAT aurora was HUGE (was seen in Mexico City). Now, Iceland is certainly a pretty good place to try to see the aurora, but there are countless places just as good or better that are easier and/or cheaper to get to for most folks (Fairbanks has not been marketed very effectively). There's no specific location that's better for seeing them than another - it's the conditions listed above - which vary every day and are impossible to predict - that determines if they can be seen.
If you want a chance to see the aurora (no guarantee): go someplace - anyplace - up north where it's very dark and very clear at night, away from city light pollution. Be outside and stay outside in the late night hours (dress warm!). Be there for a while - ideally for weeks or months. The longer you stay the better your chances. If you can't be there for weeks or months, then you will need to be lucky.
Iceland is an interesting place, and it has certainly become a trendy tourist destination in recent years. I was there about 30 years ago when it was anything but trendy. I liked it. But I saw no aurora while I was there (in November).
Hope this helps. Good luck!