We know her best for sculptures and sculptural installations of an almost overbearing ferocity and eeriness: spiders on the scale of giants; nasty, confining, cell-like structures with their horrible, thrift-shop furnishings; or those watchtowers with looming mirrors which menacingly oversaw the opening of Tate Modern's Turbine Hall in that apocalyptic year of 2000.

But there was a side to Louise Bourgeois, who died on 31 May of this year at the age of 98, that was much more yielding. Fabric, that malleable stuff, was passing through her fingers, on and off, for more than 80 years. She first learnt to use needle and thread when she was 12 years old, and had to repair ancient tapestries in the family workshop at Aubusson. Old tapestries tend to get ragged at the bottom, so her speciality came to be the replication of missing hands and feet. She became quite deft at drawing and then replacing those human appendages. Clothes were enormously important to her too – the memories they stored; the lives they'd lived. She often incorporated them into her work, in whole or in part. And it is these small-scale fabric pieces made from samples of materials hoarded over a lifetime (dresses, table napkins, shirts, tablecloths, face cloths), on which she was working until the last two weeks of her life, that we can see in this final show.

The great sea-change in her life occurred in 1938, when she moved to America and married an art historian called Robert Goldwater. Thereafter she made few two-dimensional works for the greater part of her later life – it was Fernand Léger who encouraged her to become a sculptor – and she only returned to them in her last decade, when travelling from Manhattan to her vast studio in Brooklyn became too much of an effort for her. What do these late works tell us about this extraordinary woman, who came so relatively late in her life to recognition?

Death is a brutal encounter at the best of times. Some artists seem to thunder towards it, working on and on with undiminished vigour. A reckless new looseness sets in. Childhood impulses of self-abandonment seem to reassert themselves. Think of Picasso or Titian – or de Kooning – a friend of Bourgeois' – who, even when suffering from the later stages of dementia, continued to work in his studio. It was as if his hands still knew what to do. Did Bourgeois work in this way? No, not really. She remained her own perversely individual self to the end, rigorously in control.

These late works don't find her raging against the dying of the light. Quite the opposite, in fact. This slowing down and removal from her studio, this enforced domestication, seem to have turned her late work into something more rationally considered, more delicately cerebral than much of the work of her middle period. It was during her 70s and 80s that she seemed to be raging and raging as she plundered the depths of her own subconscious, smelling the burning. In extreme old age she seems more reconciled to her own lifelong practice. There is a new gracefulness, a new calmness, a new harmoniousness to the work. She seems as if she wants to effect some kind of reconciliation with herself, and with the demons that plagued her. She is almost standing apart from herself. We no longer feel that she is making art to save herself. These late pieces, which are made from fabric scavenged from bed linen, tablecloths, napkins, dresses, shirts, feel relatively free of trauma.

Bourgeois, in these late works, explores old themes in refreshingly different ways. The sleek new Hauser & Wirth space in Savile Row has two galleries, north and south. The theme of spiders and spirallings is addressed in the south gallery. The single most touching work – and that is not a word one would readily associate with Bourgeois' name – in the entire show hangs in the south gallery, just beside the reception desk. It is a spider, made in 2007, from multi-coloured fabrics, delicate fragments of cloth, woven under-and-over, over-and-under. The body of the beast, gently pulped up, would just about sit in the palm of a human hand. Framed and sealed behind glass, it is attached to a small, blue-bordered kerchief, and it is fabricated from the most delicate ribbonings of pink, blue, grey and yellow. Its eight legs seem to squirm out from its body in all directions, like scribbling on the air. It is both comic and delightfully joyous.

Just feet away from this fabricated arachnid sits another of its kind, from 2003. It is called "Crouching Spider", and it occupies a different psychological terrain altogether. This is a Bourgeois spider of menace, perhaps 20 feet in diameter. Its long, steel legs of glossy black steel with their weird articulations slither out from its pendant middle. Its blackness sings out at us – resented authority, as Bourgeois once described the nature of blackness. A bold child might choose to creep underneath it, very fast.

There have been many spiders in the life of Louise Bourgeois. She has interpreted them for us. She has raised them to the level of symbols. The spider is the mother – her own beloved mother – because she is a weaver, a spinner, a repairer, a maker. And an unmaker. This giant is a protectress too, but she is a frightening presence because she needs to repel whatever it is that might threaten to emerge from the darkness. That is what we feel about these giant spiders, that they need to outfrighten the frighteners.

The atmosphere of this small, wall-mounted spider on the other hand, made from colourful fabrics, is altogether different. Bourgeois is admiring it for its beauty, its dexterity; she is celebrating it for being such a touchingly brilliant construct, for its evolutionary triumph: is it not a piece of unparalleled machinery? Its shadow as menacing symbol has dissolved into the air.

And so it is with many of these fabric works. They are so often celebrations of one kind or another, unburdened by the darkness of her autobiography. They often remind us that one of her early passions was mathematics – and geometry in particular. These playfully intricate works – and, yes, there is more than a touch of guileless play here too – seem, in part, to be a celebration of the beauty and the sheer appropriateness and inevitability of geometrical forms.

In several of these fabric collages, she begins with the geometrical perfection of the spider's web, and then she replicates that form, over and over, placing one web form beside another, interlocking them. At the centre of each of these webs, she often places a small fabric flower. Sometimes it is a pink rose. Sometimes the ribbings of the web stand proud of their surfaces. When placed side by side in this way, these web forms dissolve into abstract patterning. Once again, the haunting symbolic force of the spider has been smoothed away in the interest of something lighter and more immediately appealing: elegant patterning. It is as if she is taking the idea of a spider and its web to pieces in order to relish it as a visual phenomenon. Another brilliantly colourful untitled piece consists entirely of what look like the tracings of the gawky movements of a spider's legs, that way it has of seeming about to move in many directions at once, or of moving and then drawing back as if each leg is a kind of experiment in movement. Other fabric pieces are homages to the ceaseless movements of the spiral. "The spiral", she once confided to her notebook, "is an attempt at controlling the chaos."

These pieces seem to exist outside the time of her own ceaselessly fraught autobiography. They feel like states of meditative detachment, beyond the stress and the anxiety of the day-to-day.

One of these spirals hangs on the wall near a sculptural installation of 1992 called "Bullet Hole". The installation consists of an encirclement of scavenged metal doors. Inside those metal doors sit three huge wooden balls. They feel much too large for the space inside which they are pent. One is cracking open. The theme of the piece is fear, confinement. In fact, there is a slogan written in metal lettering on one of those doors: "Fear Makes the World Go Round".

By contrast, the late pieces on the walls, made a decade and a half later, are giving us an utterly different massage. It feels like a message of reconciliation. And perhaps the patient, painstaking act of sewing has helped to bring this about. "I always had the fear of being separated and abandoned", Louise Bourgeois once wrote. "The sewing is my attempt to keep things together and make things whole."

Louise Bourgeois: The Fabric Works, Hauser & Wirth, London W1 (020 7287 2300) 15 October to 18 December