In Game of Thrones S8E5, Daenerys goes on full killing mode and destroys the entire city in spite of the bells ringing, indicating Cersei surrendering. (It's evident everyone knows the bells ringing means surrender, including Dany.) But even after Cersei and King's Landing surrendering, why did Dany go ahead and destroy the entire city?

There is grief, and then there is rage, and then there is madness. We have seen her handle grief and anger almost always well. The only times she's unreasonably ruthless is when she kills the Tarlys.

So why did Dany do this in spite of all the pep talk from all her advisers and the bells ringing?

She has become mad pragmatic

While some will argue she has become as mad as her father, I will argue she has all her mind.

One important element to understand this is the dialogue she had with Jon Snow before the attack

Far more people in Westeros love you than love me. I don't have love here. I only have fear [...] Alright then," she says. "Let it be fear.

She knows she won't earn love from his people. And as Machiavelli says:

From this arises the question whether it is better to be loved rather than feared, or feared rather than loved. It might perhaps be answered that we should wish to be both: but since love and fear can hardly exist together, if we must choose between them, it is far safer to be feared than loved
^{the prince (1513)}

As Benioff in the "inside the episode":

I think that when she says "let it
be fear" she's resigning herself to the
fact that she may have to get things
done in a way that isn't pleasant and
she may have to get things done in a way
that is horrible to lots of people.

Why did she react like this to the bells?

However, she is not totally emotionless, and it's true that there is rage inside her when the bells tolls. Because at this moment, she realised that she could have easily take Kings Landing, two seasons ago.

She realized she could have dealt with the Night King threat after becoming Queen.

She lost two dragons, Jorah, Missandei, and a big part of her army, while a simple attack with only one dragon was enough.

She realized all those previous sacrifices were useless.

How can she be such a good ruler in Essos, but so cruel in Westeros?

While it seems odd that the same character rule in a quite different way between two realms, this is not unrealistic. Bruce Bueno de Mesquita, political scientist and game theorist, studied the reign of Leopold II, King of Belgium and Congo. The exact same person rules these two kingdoms in very different ways, but there is a reason for this: the power was not held by the same people/structure. Two different kingdoms, two different ways to conquer/keep the power.

Abstract

From 1885 until 1908 Leopold II was not only the King of Belgium but also the personal owner of the Congo Free State. The policy outcomes during his reign turned out to be fundamentally different in the two countries: Whereas in Belgium he improved living conditions, in the Congo he established a brutal tyranny. This paper analyses the reasons for these different leadership styles of Leopold II by means of the 'selectorate theory'. The selectorate theory explains policy outcomes as a function of governance institutions. It assumes that the ruler maximizes his own utility which means first of all to sustain himself in power. Under Belgium's governmental institutions Leopold II required broad support from the general public but in the Congo he only needed a very small group of supporters. To reduce the possibility that Leopold's different leadership styles were caused mainly by racism his period is compared to the reign of the Congolese leader Mobutu Sese Seko.
^{Leopold II and the Selectorate: An Account in Contrast to a Racial Explanation, Bruce Bueno de Mesquita}

The same thing can be said for Essos and Westeros. In Essos, she is seen as a liberator and can earn the love of her people, whereas she is seen as a conqueror in Westeros, and must rule with fear

Like father like daughter: Mad.

It's been hinted before too when she burned Dickon Tarly and his father despite Tyrion's attempts to suggest them to put in cells. Also been hinted by her urge to sit in Iron Throne even when she knows Jon is a contender too and will be fair to say more deserving.

She had seen so many losses Husband, unborn son, two dragons, so many Dothraki and unsullied but the loss of Missandei was the last nail in the coffin and she did seem to lose her calm.

Also, remember what Missandei said before dying:

Dracarys

And that's what she did. Also, she was ready to do this before too but Tyrion/Jon, etc stopped her.

Marie Claire tried to bring all the clue/hints that she was always destined to be the mad queen like his father as he was not mad from start either:

• 
  

  When she snaps at her evil brother, Viserys, in Season 1, episode 4.

  
  
  

  
  

  "The next time you raise a hand to me will be the last time you have hands."

  
  

  
  



• When Khal Drogo kills Viserys, in Season 1, and she is eerily calm about it.



• 
  

  When she marches on Qarth at the beginning of Season 2 and threatens to "burn cities to the ground."

  
  
  

  
  

  "When my dragons are grown, we will take back what was stolen from me and destroy those who have wronged me. We will lay waste to armies and burn cities to the ground."

  
  

  
  



• When the merchants in Qarth won't give her ships, she announces she will take what she believes is hers with "fire and blood."




• When she feeds one of her enemies in Meereen to her dragons.




• When she decides to burn the Khals to death in Season 6.




• When she burns the Tarly men to death in Season 7. ( which I already mentioned before in answer)

She was on the brink.

At this point, she is so alone and without many of her long term friends that she is overcome with the isolation and rage that accompanies it.

David Benioff: Dany is an incredibly strong person. She's also someone who has had really close friendships and close advisors for her entire run of the show. You look at these people who have been closest to her for such a long time and almost all of them have either turned on her or died, and she's very much alone. And that's a dangerous thing for someone who's got so much power, to feel that isolated. So at the very time when she needs guidance and those kind of close friendships and advice the most, everyone's gone.

So, she is all alone and seems that she simply doesn't care anymore. Most of her friends are dead, she thinks Jon betrayed her and still doesn't have the throne. So, she throws away her signature compassion and takes the throne with Fire and Blood.

Realpolitik

Although the producers specifically showed Dany having an emotional breakdown, and stated that she makes it personal (re: Cersei's treachery and resistance) there is also a practical political intent:

• Dany demonstrates the destructive power of even a single dragon, and sends an unambiguous message that any who resist her rule face utter destruction.