I would say that Lear achieves wisdom by truly understanding his own predicament and his own self of self in the middle of that awful storm. Prior to that point, Lear had believed that his own wealth had meant value. He believed that with title, property, possessions, and wealth, he was able to externally reflect the sense of respect and dignity that he believed he held and to which he was entitled. When asking his girls to profess their loyalty, he believes Regan and Goneril, primarily because like all else he possesses this is an external representation of his internal hopes. When he walks into the storm and understands that when these external elements are stripped, there has to be a rumination upon what remains and what is internal. When Lear understands this, it is as this point where wisdom, what the Greeks would call "sophia," is present.
The journey of the old king in Shakespeare's immortal tragedy, King Lear, is a journey from folly to wisdom through madness and suffering.
Lear is old and impassioned, sometimes, even childish. The way he declares to divide & distribute his kingdom among his three daughters on the basis of a sort of elocution contest, the way he banishes his most loving daughter, Cordelia, without understanding that her 'nothing', however obstinate it sounds like, is everything, we may agree with Kent that the anger-driven octogenarian must be 'mad' from the outset. Irascible anger clouds Lear's judgement. By banishing Cordelia and by submitting himself to the custody of Goneril & Regan, the two 'pelican daughters', he foolishly banishes true love and embraces the hypocritical appearances of it.
Now let us trace Lear's passage from this utter folly to his attainment of final wisdom in the recovery of Cordelia's filial love and mad Lear's regeneration, tragic reversal in her murder, the closing moments of his further degeneration(?) & death:
a) Lear's persecution by Goneril & Regan--driven out onto the open heath, unsheltered & exposed to the cruelties of natue and cosmos;
b) Lear runs about in tattered clothes, accompanied by his Fool and Edgar diguised as a Bedlamite--feels close affinity with the 'poor naked wretches' of the world---the folly-fallen old king empathises with all the sufferers of the world;
c) This process of learning through sufferings is enhanced by the constant commentaries by the Fool, functioning as the old king's alter-self and his conscience; the Fool tells Lear that the king is a greater fool;
d) As Cordelia returns to rescue her wronged father, Lear retrieves his lost love that has a healing effect upon his degenerate mind; Cordelia is a sort of Christ figure offering Lear the road to salvation;
e) The loyal Kent & the good and humane Edgar also contribute to Lear's journey to wisdom; both of them illustrate sympathy and support necessary for Lear's recovery from madness.
King Lear thus progresses through 'filial ingratitude' to filial love, through enormous physical as well as mental sufferings to the brink of sanity and realisation, though his story ends on a final impression of disaster.