Littlefinger tries to kill Tyrion

Who actually sent the assassin to kill Bran?

It was Joffrey. Tyrion and Jaime found out for themselves.

Tyrion was already suspicious of Bran's fall.

Myrcella gasped happily and Tommen smiled nervously, but it wasn't the children Tyrion was watching. The look between Jaime and Cersei lasted no more than a second, but he didn't miss it. Then his sister let her gaze fall on the table. “This is no mercy. These northern gods are cruel to make the child abide in such pain. "
AGOT - Tyrion I.

And there was something Joffrey said about killing Bran's Wolf.

Tyrion looked down and saw the dog standing with young Joffrey as squires swarmed around them. "At least he dies quietly," replied the prince. "It's the wolf that makes the noise. I could barely sleep last night."

Clegane cast a long shadow across the hard earth as his squire lowered the black oar over his head. "I could silence the creature if you like," he said through his open visor. His boy put a long sword in his hand. He tested the weight and cut into the cold morning air. Behind him the courtyard rang between steel and steel.

The prince seemed to enjoy the idea. "Send a dog to kill a dog!" he exclaimed. "Winterfell is so infested with wolves that the Starks would never miss one."

Tyrion jumped into the courtyard from the last step. "I disagree, nephew," he said. "The Starks can count after six. Unlike some princes I could name."
AGOT - Tyrion I.

But of course that meant nothing more than to give a good idea that Joffrey was capable of murdering something for his entertainment.

Then Catelyn arrested him for conspiracy to murder Bran. Tyrion was innocent, but he knew his siblings had something to do with the attempt, although he knew them well enough to suspect that the second attempt might not be a product of Cersei's mind or Jaime's hand, due to the sheer clumsiness of the idea had been, but he didn't completely rule out the possibility.

His sister was not without a certain cunning, but her pride blinded her. She would see the insult in this, not the opportunity. And Jaime was worse, careless and stubborn and angry easily. His brother never untied a knot when he cut it in two with his sword.

He wondered which of them had sent the footpad to silence the Stark boy , and whether they really had conspired when Lord Arryn died. When the old hand was murdered, it was done skillfully and subtly. Men his age died from sudden illness all the time. In contrast, it seemed incredibly awkward to send a fool with a stolen knife after Brandon Stark found him awkward.And wasn't that strange, come and think about it ...
AGOT - Tyrion V.

Then he heard all about a particular knife and how Littlefinger had framed it.

It wasn't until Joffrey's wedding that Tyrion put the pieces together.

"Take care, Your Grace," Ser Addam Marbrand warned the King. " Valyrian steel is dangerously sharp.

" I remember . "Joffrey put Widow's Wail in a wild two-handed disk on the book Tyrion had given him. The heavy leather cover parted in one fell swoop. Sharp!I told you I am no stranger to Valyrian steel.

It took half a dozen more cuts to hack apart the thick book, and the boy was breathless when he finished. Sansa could feel her husband struggling with his anger when Ser Osmund Kettleblack exclaimed, "I pray that you will never turn that wicked edge on me, sir." "See you never give me a reason, ser." Joffrey tossed a piece of the life of the four kings off the table with the point of his sword, then shoved Widow's Howl back into its sheath.

"Your mercy," said Ser Garlan Tyrell. “Maybe you didn't know. There were only four copies of this book in all of Westeros, illuminated in Kaeth's own hand. "

"Now there are three." Joffrey undid his old sword belt to put on his new one. " You and Lady Sansa owe me a better present, Uncle Imp . This one is all hacked into pieces. "

Tyrion stared at his nephew with his mismatched eyes. " Maybe a knife, sir.To bring your sword together.A dagger made of the same fine Valyrian steel ... with a dragon bone handle, shall we say?

Joff looked at him sharply."You ... yes, a dagger that goes with my sword , good. "He nodded." A ... a golden handle with rubies. Dragonbone is too simple. "
ASOS - Sansa IV

Joffrey's reaction confirmed Tyrion's doubts in his head.

I am no stranger to Valyrian steel, the boy had boasted. The Septons have always been about how the Father above judges us all. If Father were good enough to knock down and crush Joff like a dung beetle, I could even believe it.

He should have seen it a long time ago.Jaime would never send another man to kill him, and Cersei was too smart to use a knife that could be traced back to her, but Joff, arrogant malicious stupid little guy he was ...

He remembered a cold morning descending the steep outside stairs of Winterfell's library and Prince Joffrey joked with the dog for killing wolves . Send a dog to kill a wolf, he said. Self Joffrey wasn't so stupid as to order Sandor Clegane to kill a son of Eddard Stark ; The dog would have gone to Cersei. Instead found the boy his cat's paw among the unsavory freeriders, merchants, and camp followers who had joined the king's party on their way north . A poxy sucker willing to risk his life for a prince's favor and a small coin. Tyrion wondered whose idea it had been to wait for Robert to leave Winterfell before opening Bran's throat. Joffs prefer it. No doubt he thought it was the height of the ruse.

Tyrion seemed to remember that the prince's dagger had a jeweled pommel and inlaid gold work on the blade. At least Joff hadn't been stupid enough to use that. Instead, he poked around at his father's guns. Robert Baratheon was a man of negligent generosity and would have given his son any dagger he wanted ... but Tyrion suspected the boy had just taken it. Robert had come to Winterfell with a long tail of knights and followers, a huge wheelhouse, and a baggage train. No doubt a hardworking servant had seen the king's weapons go with him in case he should covet any of them. The blade Joff had chosen was beautiful and simple. No gold work, no jewels in the handle, no silver inlay on the blade. King Robert never wore it, probably forgetting he had it. But the Valyrian steel was deadly sharp ... sharp enough to cut through skin, flesh, and muscle with one swift blow. I am no stranger to Valyrian steel.But it was him, wasn't it?Otherwise he would never have been stupid enough to pick Littlefinger's knife.
ASOS - Tyrion VIII

Jaime found out for himself too, believing Joffrey did it to impress his father.

Jaime suddenly remembered something else that bothered him about Winterfell. " At Riverrun, Catelyn Stark seemed convinced I had sent a foot pad to cut her son's throat . That I gave him a dagger. "

"That," she said scornfully. "Tyrion asked me about it."

“There was a dagger. The scars on Lady Catelyn's hands were real enough, she showed them to me. Did you... ?"

"Oh, don't be absurd." Cersei closed the window. “Yes, I hoped the boy would die. So you have Even Robert thought it was best."We kill our horses when they break a leg and our dogs when they go blind, but we are too weak to show the same grace to crippled children," he told me . At that time he was blind himself because he had had a drink. "

Robert? Jaime had guarded the king long enough to know that Robert Baratheon was saying things in his cups that he would have angrily denied the next day. " Were you alone when Robert said this?

“Don't you think he told Ned Stark, I hope? Of course we were alone.We and the children . "Cersei removed her hairnet, draped it over a bedpost, then shook out her golden locks." Maybe Myrcella sent this man with the dagger, you think so?

It was intended as a mockery, but she had hit it exactly, Jaime saw immediately. " Not Myrcella.Joffrey . "

Cersei frowned. “Joffrey had no love for Robb Stark, but the younger boy was not for him. He was just a child himself . “

A kid who's hungry for a pat on the head that you make him believe is his father. He had an uncomfortable thought. “Tyrion almost died from that bloody dagger. If he knew the whole thing was Joffrey's job, that could be the reason ... "
ASOS - Jaime IX

Citadel also takes note of this:

The evidence revealed in A Storm of Swords directly points to Joffrey as the instigator. At the Queen's breakfast he revealed: "... I am no stranger to Valyrian steel." (III: 663) Tyrion, suspicious of this statement, suggests giving Joffrey a Valyrian steel dagger with a dragon bone handle as a gift. Joffrey's response is uncomfortable: "Joff gave him a sharp look. 'You ... yes, a dagger that goes with my sword, good.' He nodded. "A ... a gold handle with rubies. Dragonbone is too plain." "(III: 664) Finally, Jaime asks Cersei about the attempt to attack Bran. She reveals that Robert said in Joffrey's hearing that it would be a kindness to end the boy's misery, but that they were all too weak. Jaime concludes that Joffrey, seeking his father's attention, may have chosen to prove that he was capable (III: 823).
The Citadel: FAQ - 5.1 Who Tried to Kill Bran?