1. Joffrey was a spoilt, cowardly and a vile person. He was also a bastard born out of incest. He inherited the evil characters of Cersei and was even evil than her. These are mainly because of Cersei’s upbringing. He was already bad and Cersei made it worse. But assuming he was a kind-hearted king, not having the sort of vile cruelty or hatred from his mother, Ned Stark would not have died in the first place. Joffrey would not have killed Ned and placed his head in a spear on the wall for Sansa to see.
He wouldn’t have made Sansa suffer and would have married her. He would have ruled wisely but his kind-heartedness would have made him weak in front of his mother and he would have acted according to Cersei’s wishes. The first thing to consider is how would Ned have reacted when he found out about Cersei and Jaime and the fact that Joffrey, being a kind hearted person, was born out of incest. He might have reacted differently towards the issue. This would have eliminated beheading Ned Stark. Also, both Sansa and Cersei pleaded with Joffrey not to kill Ned.
Cersei knew that great consequences would happen if Ned Stark dies. But it was Joffrey who ordered the execution. If he had been a kind king he would have spared Ned’s life and sent him away to Winterfell
2. Even with the influence of Tyrion and Cersei, Joffrey would have acted according to his own will like Tywin, if he were a kind hearted king. Hence, the brutal murders at the Red Wedding would not have happened as Joffrey would not have allowed it to happen. In the first place, he would not have been killed by Olenna Tyrell and Petyr Baelish.
Joffrey would have been made to cast Sansa aside and marry Margery Tyrell so as to have a more secure bond between the two kingdoms. So probably Joffrey would not have died during his wedding. When the Faith takes Margery and Loras, Joffrey would have done the same thing as Tommen. Tommen was a kind hearted king and did things that he felt was right, even though he was influenced by his mother (Tywin was killed by Tyrion). Ned would have probably reacted the same way he did as his contempt towards the Lannisters started since the day he saw Jaime with his sword dripping red with blood and the body of Aerys Targaryen on the floor. He also knew about what Tywin ordered the mountain to do to Elia and her children. Hence, Ned would have talked with Cersei about the secret he learned about Joffrey. He might have been imprisoned, but certainly not been beheaded. He might have been pardoned and sent to the North or kept as a prisoner in King’s Landing itself.
3. Had Joffrey been a kind-hearted king, the advent of the events that followed him being cruel would not have changed much for the exception of Ned not being dead. But that alone would have changed much in the events that followed the Starks. Robbb would not have been killed nor would Winterfell have been captured by Theon or Ramsay.
None of the Starks would have experienced the turmoil they went through. Plus Daenerys coming over to Westeros to conquer the Iron Throne would have portrayed her as a usurper.
Joffrey being a kind hearted person would have changed many things. Nymeria would not had to be sent away by Arya because of the incident with Joffrey at the river bank. Hence, Lady would not have died. This would mean that King’s Landing would have had two Dire wolves. This would have changed the situation for both the girls in King’s Landing. A lot would have happened for the good had he been kind, but GRRM had to kill Ned, Robb and many other characters before Joffrey dies.