Two nights, all walkable from the hotel.
Around the hotel — Strøget, the Latin Quarter, Nyhavn, Kongens Nytorv
Christianshavn — canals, Christiania, Kadeau
Refshaleøen / Holmen — the post-industrial harbour
Nørrebro — the youngest, most multicultural part of town — Jægersborggade for design, plus the natural-wine bars
Vesterbro — edgy and cool; the Meatpacking District, design shops along Vesterbrogade
Østerbro — leafy and residential (Juno, Normann Copenhagen, Kastellet)
Frederiksberg — elegant and green
Designmuseum Danmark, the David Collection, the National Gallery, Ny Carlsberg Glyptotek, Rosenborg Castle, Rundetårn.
The rhythm I'd suggest: one day for nothing at all, one for a guided sea outing or a RIB safari across the Vestfjord, one for a longer hike, and one simply on and around the island.
Unstad, the Arctic surf beach. Even with no one in the water, the drive in over the pass and the bowl of the bay is one of the great Lofoten views — and there's a small surf café for coffee.
Haukland and Uttakleiv, back to back on the same headland and both improbably white-sand-and-turquoise. Uttakleiv is the photographer's; Haukland the one you can actually swim at (bracing). A short, flat coastal path links the two.
A sea outing from Ballstad — RIB safaris and fishing trips run from the harbour, and the sea eagles are near-guaranteed. Hattvika Lodge can arrange it directly, which is easiest.
Ramberg and Skagsanden — wide white-sand beaches about 40 minutes away, dramatic in any weather.
A full day down to Reine and Å: Hamnøy for the classic shot of red cabins under spiked peaks, from the bridge (best in the evening when it's quiet — park on the east side and walk up); Sakrisøy, where you stop at Anita's Sjømat, the beloved fish deli, for the legendary fishcakes and the best fish burger in Lofoten, eaten on the dock — the best casual lunch in the south; Reine, the postcard village itself; and Reinebringen, the famous climb — steep stone steps to an outrageous view over Reinefjorden (start early, skip it if it's wet, 2–3 hours return). Å is literally the end of the road and the last letter of the Norwegian alphabet — the southernmost village in Lofoten.
A short drive east.
Festvågtind — the classic short climb (around 1.5–2 hours return) with a postcard view straight down over the village and its famous football pitch. The trailhead is just outside town.
KaviarFactory — a contemporary art gallery in a former roe factory, genuinely good and exactly your kind of place.
Summer snorkelling — Hattvika and the Henningsvær operators can set it up.