In which I summarize that more cultural aspects of Iceland, beyonds its natural beauty, ugly rock strewn fields, and thermal marvels.
Naturally, most of the museums are in Reykjavik. Lets face it, the ONLY city in Iceland worth seeing. And there is not that much territory to cover, so it will not take you long to create your sightseeing itinerary.
There is the main shopping area. Laugavagur street. On this street you can find, approximately 50 souvenir shops that sell exactly the same Viking and Puffin theme gimcracks, gewgaws, and t shirts. Yawn. I bought a shotglass, which my wife and I always do. The rest is, like most souvenirs world wide, not anything anyone actually needs. This main shopping street is 1.2 miles long, approximately one aisle at a Macy’s. You can do it an hour or so, if you also stop for coffee.
To make sure you are congnizent of the Iceland support for LGBQT++++ movement, they painted a big rainbow on Laugavagur street. Virtue signal-check!
At the top of the street is a gorgeous church, Hallgrimskirkja, (pronounced in ancient Norse/Icelandic “hrdudumfunder”.) Its beautiful from the outside, meh on the inside. The Lutherans are not much for displays adornment inside their churches, being a sober and often depressed bunch. Pretty sober bunch, those Lutherans. Spectatular church though.

Hallgrimskirkja Lutheran Church.
Reykjavic also has a really cool whale museum, wherein you can see lifesized versions of many whale species. Well, some scaled down a bit, to fit inside. This we really enjoyed.
Also available to see is the famous penis museum, in which you can see artificially stiffened penises of many species on display. This is the MOST VISITED museum in Iceland. Especially populur with Covette owners, I hear. Really! Its every guys nightmare, every guy I saw there walked around with his legs crossed and a pained expression on their faces. Women, on the other hand, were delighted. The thought of doing a Lorena Bobbit on their husbands, boyfriends, pet doberman, seemed to brighten their days. Make of this what you will. The museum is accepting donations of exhibits, it is recommended that you make yours poshumously.

Woman in Phalliological Museum. Happy to be holding a really big one!
Last on our visit agenda was a tourist trap, a whole room made of ice with cool viking ice sculptures. Called the Ice bar. I recommend this. You get a free drink with admission, and get to sit in an ice throne. Its hokey but fun.

Well thats it, a snarky but affectionate review of the sights of Iceland. Five days, including transit to and from, is plenty unless you are a camper or want to explore the 99% of the territory where no one lives. You will need to get to the airport 3 hours early, as the security to leave takes FOREVER. There are 3 separate checkpoints, with long lines at each. Can’t explain this.
BTW the Dublin airport is FAR WORSE. Give yourself five hours to get through American customs there.