Seeking advice: Has anyone here been to Iceland?

I’m going to the Arctic for 12 days in early July: Svalbard (formerly called “Spitzbergen”) Jan Mayen, and Iceland.  This is a cruise, and it’s on my own dime as I guess I’m getting to old to be asked to lecture for Hurtigruten (this is a Quark trip).  I fly from Chicago to Helsinki, from where we fly the next day to the world’s northernmost town having more than a thousand people, Longyearbyen in Svalbard, an archipelago owned by Norway.  Here are some unique aspects of the town from Wikipedia (bolding is mine):

Due to its remoteness, Longyearbyen has laws that are found in few, if any, other places in the world. Notable examples of such laws include a ban on cats, a restriction on the amount of alcohol an individual can purchase each month, and a requirement that any individuals venturing outside carry a rifle for protection against polar bears. While it is popularly claimed that it is illegal to die in Longyearbyen, the wording of this claim is misleading. While it is not actually illegal to die in the town, there are no options for burial of bodies there (ashes can be buried with permission from the government) and residents considered terminally ill are typically required to move to the mainland. The decision to disallow burials came in 1950, when it was discovered that the bodies of residents who had died as a result of the 1918 flu pandemic had not begun to decompose. Today, scientists are concerned that these corpses, preserved by permafrost may still harbor live strains of the virus responsible for killing between 1% and 6% of the world’s population during the early 20th century.

No cats!!!!

At any rate, then we go cruising, looking at the scenery at Jan Mayen and hoping to see polar bears, whales, and other Arctic marine life.  After ten days or so we land in Reykjavík, the capital of Iceland. I believe most of my fellow passengers will be flying home the day after we dock there, but, given that I’m unlikely to make it back to Iceland before I croak, I wanted to stay a few extra days to see a country that’s supposed to be spectacular.

If you have been to Iceland, I’d appreciate any tips you have for me.  I was planning on staying four or five extra days because I can’t linger indefinitely. Is that long enough? What things are good to see? Any recommendtions for food or lodging (I don’t like fish)?  Please put advice in the comments. Thanks!

Here’s Longyearbyen (pop. 2,295) with the caption, “These are all the buildings of this kind (“spisshusene”) left standing after the avalanche disaster in 2015.”

Longyearbyen and the location of Svalbard. Greenland and iceland are visible to the west and southwest:

Location of Jan Mayen. Some info from Wikipedia:

Jan Mayen Island has one exploitable natural resource, gravel, from a site located at Trongskaret. Other than this, economic activity is limited to providing services for employees of Norway’s radio communications and meteorological stations located on the island. Jan Mayen has one unpaved airstrip, Jan Mayensfield, which is about 1,585 m (5,200 ft) long. The 124.1 km (77.1 mi) coast has no ports or harbors, only offshore anchorages.

. . . . The only inhabitants on the island are personnel working for the Norwegian Armed Forces and the Norwegian Meteorological Institute. Eighteen people spend the winter on the island, but the population may roughly double (35) during the summer, when heavy maintenance is performed. Personnel serve either six months or one year and are exchanged twice a year in April and October.