How did David's struggle come to an end?

The lovable old eccentric Miss Betsey Trotwood is the answer to David's prayers. His great aunt provides him with a haven of peace and security just when he needs it most. Up until now, poor David's had a pretty miserable life. His wicked stepfather Mr. Murdstone and his evil relatives have taken over the Copperfield house, subjecting David's mom to an unimaginable domestic tyranny which undoubtedly contributed in no small measure to her untimely death.
As for David himself, he's packed off to Murdstone's blacking factory, where he toils away in horrendous conditions, sticking tables on bottles and packing them in boxes. It's a truly wretched existence; no wonder David goes AWOL at the first opportunity.
And no wonder that he seeks sanctuary in the welcoming bosom of Betsey Trotwood. Though she gives the impression of being a bit of a scatter-brain, Betsey's a remarkably practical woman, and as well as loving care, she provides David with some much-needed guidance in his life. She's responsible for sending him to school—a good one this time, not like that horrible dump run by the unspeakable Mr. Creakle.
Betsey's also thinking of David's future career prospects, which is why she pays for him to be articled, or apprenticed, to a lawyer. We might say that Betsey's a substitute mother for David except that his own mother was always too frail and too weak to take charge of her son's life like this. Betsey's like the mother that David never had, and then some.

I'm going to assume you mean "When does David finally—finally!—catch a break?" Things start looking up for David when he meets his great-aunt, Miss Betsey, and learns how to rely on himself. He has to walk from London to Dover just to find her, but she's worth every blister.
Miss Betsey protects David from his past so he can focus on the future. When the Murdstones follow him to her house, for example, Miss Betsey tells Mr. Murdstone and his equally abusive sister to hit the bricks. She's charitable, but she's tough.
She also helps David secure the tools necessary for success on his own terms; all he has to do is provide the hard work. Miss Betsey sends David to a good school, and she pays for his law internship in London. Because of Miss Betsey's generosity, David is ultimately able to take care of not just himself, but also his family—something his own mother was unable to do for him.