My espresso is still hot. I can feel it through the cup, held in both hands, which is more than I will be able to say for it in an hour, once the photographs have pulled me under. This is the best hour of the day, the one before the city is fully awake, when yesterday is close enough to touch. We are in the courtyard of the Stockholm Stadshotell, on the island of Södermalm. The smell of coffee and warm stone. The first sun of the day on the back of my neck, a bicycle ticking past beyond the wall, and the light doing something to the old brick that it flatly refused to do two days ago.

Two days ago, it rained.

Not romantically. It rained the way it rains when an umbrella becomes a matter of opinion rather than protection, when the wind comes up off the water and turns the whole contraption inside out. You could hear it: rain hissing on the harbour, halyards ringing against their masts, the hush of tyres on wet cobbles. The whole city smelled of cold lake and wet stone. Sensible Stockholmers folded themselves into cafés to wait it out; we did not. We walked, water running off our cuffs and down the backs of our necks, past quays the colour of the sky, laughing at how thoroughly we were losing. A city has to trust you a little, I thought, before it shows you its good rooms. That day, Stockholm was still making up its mind.

Yesterday, it relented.

The morning took us to Rosendals Trädgård, a garden on Djurgården, the leafy island Stockholmers treat as their communal back garden, reached across a short arm of water that, on a good day, hands you back the whole of the sky. This was the reason we had come to Sweden at all. A year ago, Lady Ashton and a cousin she had never met began writing to each other across that water, two branches of a family that split apart centuries ago, feeling along the thread that once tied them together. Email answered email. A friendship built itself on Messenger. Somewhere in it a promise took shape: one day, in person. The family chose the garden and marked a date on the calendar. Yesterday, the year ran out.

Walking up the path, gravel crunching underfoot, the scent of grass and apples rising from the beds, we are not quite the strangers the situation suggests. You brace for the awkward choreography, the careful questions, and it never comes. A chair scrapes, someone stands, and within minutes we are all talking over one another, a year of messages carrying on out loud. The talk slips between English and Swedish and back. Someone apologises for their English. Someone else apologises for their Swedish. Neither apology proves necessary. We interrupt one another with equal enthusiasm, laughing at the places where the two languages meet and politely refuse to cooperate. A hand lands on a forearm to make a point. The resemblances keep announcing themselves around the table, a chin here, the set of an eye there, and whatever distance the centuries had put between us burns off with the last of the morning damp.

At some point Lady Ashton and her cousin lean in for a photograph, then another, then several more. Before we leave, I slip my phone into my pocket without looking at them. There will be time tomorrow, I think.

I have those pictures in front of me this morning. I keep going back to them, the two of them caught mid-laugh, having known each other for twelve months and one day.

Rosendals has been farmed biodynamically for more than forty years, and most of what it grows travels only a few steps before it turns up in the café or the wood-fired bakery. You can smell all of it at once: fresh bread, cut herbs, earth still drying from the week's rain. Bees work the lavender. Under the apple trees, people sit with coffee and nowhere in particular to be.

Lunch arrives looking almost disappointing: Arctic char, the season's first potatoes, a dressing of aggressive green, not a tower or a foam or a tweezer-placed garnish in sight. Then you taste it and forgive it everything. The fish has been cooked so gently it barely concedes the point of being raw; it gives way under the fork like something reluctant to leave. The potatoes carry the earthy sweetness that belongs only to a first crop. The dressing reaches the back of the nose before the tongue, sharp and herbaceous, bright enough to make you sit up. A glass of cold beer sweats against the palm, cutlery ticks against plates, and the family stories keep coming as the afternoon loosens around us.

Then we cross the island to the Vasa Museum, and everything goes quiet.

Nothing prepares you for the first sight of the ship. You step out of the daylight into a deliberate dark, the air suddenly cool against your skin and carrying the faint scent of resin and tar from four-hundred-year-old oak. Your eyes adjust. You look up, and there she is. Vast, rising the full height of the museum like something surfacing from another century. The Vasa went down in this harbour on her maiden voyage in 1628, having sailed barely three-quarters of a mile before a gust leaned her over and the water poured in through her open gunports. She lay on the harbour floor for 333 years. When they raised her in 1961, she came up astonishingly whole.

The room holds its breath. Footsteps move softly on the wooden ramps. A little boy points towards the towering stern and whispers a question to his father, and the answer comes back just as softly, as though both understand that voices belong at a different volume here. Above it all the ship simply looms. Your neck strains as carved figures crowd the stern and great beams disappear into shadow, every surface still carrying the print of a human hand. You think of the men who built her, the sailors who boarded her that morning in their good clothes, and the moment this monument to royal ambition began, quietly and catastrophically, to fill. She never reached a battle. Nearly four hundred years on, strangers still stand beneath her and fall silent.

I found myself thinking of the garden that morning, and of a family gathered back around a table.

By evening, Lady Ashton wants the real thing, Swedish food cooked by Swedes for Swedes, so we walk to Pelikan, a Södermalm institution whose lineage runs back to a beer hall opened in 1664. Outside, the summer light is still high and level, refusing to admit it is nearly night. You push through the door into a wall of noise and heat, into the rich smell of gravy and fried onion and spilled beer, the room loud with conversation and warm enough to suggest that air conditioning was considered here and rejected as a failure of nerve.

A Negroni arrives, cold enough to bead with condensation, bitter and deeply welcome. We snap rounds of crisp Wasa bread, the crack of it sharp beneath the thumb, and wait for the meatballs, which land so quickly I begin to suspect the kitchen of witchcraft.

I let Lady Ashton take the first bite. Her face is a more reliable review than anything I could write. She cuts into a meatball, tastes it, and smiles. That is the review. They are tender and light, the lingonberries tart against them, the pickled cucumber a clean flash of sour.

I make it a rule not to complain about food. A kitchen's cooking is its own act of creativity, and I am a guest at the edge of it rather than a judge of it. I will break the rule exactly once, and only in a spirit of public service.

Should you ever find yourself at Pelikan, do not order the mushroom dumpling.

I shall say no more.

You will thank me.

We are seated, fed and back on the street inside half an hour, which, I have decided, is the highest compliment a kitchen can pay you, whatever the French may make of it.

Which brings us back to the courtyard, and the espresso, and the light.

The Stadshotell fills a handsome brick building put up between 1873 and 1875 as a charitable home, in memory of King Oscar I, to shelter elderly women of respectable family who had come down in the world. It has been restored with restraint: the old chapel is now the dining room, the bar spills into the courtyard, and the smell of hot fat and fresh coffee announces the serious business of the morning, a plate of pommes frites that may be the best I have eaten. They owe an obvious debt to the triple-cooked chips of The Fat Duck: deeply golden, shattering beneath the teeth, the centre soft as warm mashed potato. The waiter told me the variety, and that only two kitchens in Stockholm use it, and I am ashamed to say I have forgotten the name. But I remember the potato. Some are born to be boiled, some to be roasted; this one was put on earth to meet hot oil three times and arrive beside a cold beer.

There is a sauna here, and an ice bath, and something called a sensory shower. I have chosen the double espresso, which strikes me as a kinder way to bring a body back to life.

A cup meets a saucer with a small clean chime. The bicycle comes round again. Breakfast is beginning somewhere inside. My phone lies beside the cup, and I pick it up again, the way I have all morning, unable to leave the two of them be.

Two days ago the rain showed us one Stockholm, guarded and grey and worth the wait. The sun showed us another.

My espresso has gone cold. The phone is still warm in my hand.