Because my miles were on Alaska Airlines, I had to contact them to buy the IcelandAir flights we want for the return from Helsinki. We're also going to do a stopover in Reykjavik. Unfortunately, after a great deal of trouble finding a set of flights that included business/first class travel most of the way back, and going all the way through the booking process up to the point of actually paying, the site dumped me out when I tried to pay, saying that the partner carrier (IcelandAir) couldn't confirm the flights. Argh!
At Lisa's suggestion, I called Alaska Airlines. I spoke to a wonderfully patient woman in Seattle who managed to track down the four flights we need (including not only the stopover in Iceland but also an overnight layover at Seattle; the routing is actually HEL-KEF-MSP-SEA-SFO), find the business/first class seats for all legs except the last one back to SFO, and she even waived the usual $15 fee for booking through an agent on account of their web site having dumped me out before I could actually pay the first time.
It's very easy to complain about customer service, and too many companies have gone to offshore phone banks staffed by people who don't speak very good English and don't know how to do anything other than what you could do yourself on the company's web site. Alaska Airlines gets high marks in our book by having people who know how to make the system work for us, and have made our return trip from Helsinki much better because we'll be flying up front in Saga class.
The best airfares for the outbound flights are also on IcelandAir, but in this case we had to go to their web site. (I'll still get Alaska miles for booking it, though.) This only works because unlike [edit: most other] legacy carriers like United, IcelandAir doesn't charge you so much for a one-way ticket that you might as well buy a round trip and throw away the return half.
The trip planning isn't complete yet, because I still need to book intermediate hotels, including where to stay in Iceland and also the overnight stay in Seattle (caused because our arrival is after the last flight of the night to SFO, so we'll spend the night and fly back mid-day the following day), but I have hopes that this won't be quite as much of a hassle as the airline flights were. If only IHG had hotels in Iceland!