How does Lear face reality when he is betrayed by his evil daughters?

When Lear's daughters Goneril and Regan both break their promises to provide for their father's one hundred knights, his already well-established ungovernable temper causes him to rush out of doors regardless of the fact that a violent storm is brewing. The storm can be seen as symbolic of the reality of the cold, cruel world. Out in the storm, Lear is experiencing the suffering of living like a homeless man. His Fool has been there and done that. He counsels Lear to go back to his daughters and make the best deal he can with them.

O nuncle, court holy-water in a dry house is better than
this rain-water out o' door. Good nuncle, in, and ask thy
daughters' blessing: here's a night pities neither wise man
nor fool.                                             III.2

But Lear cannot bring himself to accept his daughters' terms. He has several motives for refusing to submit to them. One is his famous temper. Another is his pride. Another is his anger. Another is his shame at having allowed others to make such a fool of him. He is beginning to realize that it was not only his daughters who were flattering him in order to manipulate him, but everybody.

They flattered me like a dog; and told me I had white hairs in my beard ere the black ones were there. To say ‘ay’ and ‘no’ to every thing that I said!—‘Ay’ and ‘no’ too was no good divinity. When the
rain came to wet me once, and the wind to make me chatter;
when the thunder would not peace at my bidding; there I
found 'em, there I smelt 'em out. Go to, they are not men o'
their words: they told me I was every thing; 'tis a lie, I am
not ague-proof.                                               IV.6

Lear has to face reality for the first time in his life. He is living practically like a wild animal. He sees clearly how money, possessions, and power mean everything in this world. He feels his own helplessness and insignificance. It seems that Shakespeare must have decided to have Lear go completely mad, because this would explain why the old king doesn't finally acknowledge defeat and go back to live with his daughters without his entourage of a hundred knights. His daughters don't hate him. They just don't care anything about him. They would be glad enough to provide him with food, shelter, clothing, and a warm bed, because it doesn't look very good for them to be sharing their father's kingdom while the wretched old man is sleeping on the ground and eating mice!

Even in his madness Lear is facing reality because he can't escape from it. In Act IV, Scene 6, Lear explains to the blind Gloucester everything that he has learned about reality, including this:

The usurer hangs the cozener.
Through tattered clothes small vices do appear;
Robes and furred gowns hide all. Plate sin with gold,
And the strong lance of justice hurtless breaks;
Arm it in rags, a pygmy's straw does pierce it.