As a result of being completely consumed by the Dark Side, he couldn't sense Luke since the latter was of the Light Side. Palpatine sensed a disturbance in The Force in Empire Strikes Back, and identified its source at the time, but by 1983 George Lucas was more concerned with merchandise sales than gaping plot holes.
Within Vader still resides a part of his former self, Anakin Skywalker, and it could be argued that this is why he can sense Luke while the Emperor cannot.
He didn't sensed Luke because as Luke grew more powerful in the Force, in the time he had with Yoda plus the one year of training on his own, had created blind spots in the vision of Dark Side users.
in universe: Vader wasn't able to sense Luke until he tapped into the Force, so the same could be true for him and Leia. Another reason is that Vader simply wasn't looking. He thought his child was dead, and then found Luke. The thought of twins never crossed his mind.
Luke : Your overconfidence is your weakness. The Emperor : Your faith in your friends is yours!
Deceased. Luke Castellan: His fatal flaw was his excessive wrath. He was unable to forgive his father Hermes for what he had been through. Luke's anger blinded him so much that he went against his closest friends.
Palpatine admits to Darth Vader that the Jedi he actually feared most was none other than Jocasta Nu.
Leia was not using the Force or was currently Force-sensitive during her confrontation with Vader. She began training as a Jedi years later. All these reasons prove my point: Vader didn't know Leia was his daughter, and felt no presence of the Force in her, as she was not currently strong with the Force.
And with how little training Luke had received, compared to Vader's years of experience, the fight is over surprisingly quick. However, there are a number of reasons the fight went the way it did. Perhaps the most important of all is the conflict within Vader, as he had just had a chat with his son on Endor.
Why didn't Darth Vader recognise Leia as his daughter in Star Wars: Episode IV? He didn't recognize her because he didn't know his daughter was alive, and he couldn't tell it from reading her mind because she wasn't aware of it herself.
Unfortunately for the galaxy, as Palpatine grew in power the shroud of the dark side fell over the galaxy, diminishing the Jedi's ability to sense the Force. Thus Yoda failed to realize how close to home the Sith really were - until it was far too late.
Leia's journey yields a time dilation of 62.6 days; however Luke experiences a time dilation of 700.8 days (1.92 years). The students concluded that Luke is therefore 1.75 years younger than Leia, possibly rendering them the first twins ever to have over a year between their ages.
Now that she knew the truth about her parentage and her Force powers, Leia trained as a Jedi with Luke, honing her Force abilities and constructing a lightsaber. But she ended her apprenticeship after having a vision that her Jedi path would end with the death of her son.
One could theorize that he bore a child through some sort of space artificial insemination, or something closer to what we've seen in canon Star Wars material. In the canon comic Darth Vader No. 25, it is revealed that Palpatine somehow used the Force to manipulate midichlorions to impregnate Shmi Skywalker.
After his ascent to the position of Emperor, Palpatine dropped the charade and everyone saw him for what he was. His psychopathy was also prevalent even prior to becoming a Sith lord, evidenced by his delinquent behavior at an adolescent age, and later his committing manslaughter on two pedestrians via a speeder.
Darth Vader only learned he had a daughter in Return of the Jedi - but there's no evidence he realized she was Leia before his death and redemption. Darth Vader may have never known Princess Leia was his daughter in Star Wars and never cultivated a relationship with her, even when he was a Force ghost.
In Star Wars Canon, only Luke Skywalker can be acknowledged as the one who truly defeated Darth Vader.
The space station was destroyed, but Vader survived, and he flew into the far reaches of space. After the destruction of the Death Star, Darth Vader became obsessed with finding Luke Skywalker, who he discovered was his son.
Thanks to Obi-Wan Kenobi having Anakin use the same move that Luke would use against him in Return of the Jedi, it's now possible to conclude that Darth Vader let Luke win by not taking advantage of the opening he knew would exist.
Theory 1: Leia Doesn't Remember Ben Is Obi-Wan
Throughout her encounters with Kenobi in the series, he refers to himself as Ben. The most straightforward explanation would be that Leia knows the person who rescued her is Ben, and never connected the dots that Ben is actually Obi-Wan.
Why Qui-Gon Jinn Was Palpatine's Greatest Threat. Unlike Yoda and the Jedi High Council, Qui-Gon Jinn was only concerned about the will of the Force. Therefore, Palpatine needed Qui-Gonn out of the way because he was better connected to the Force than the other Jedi.
The reason why Qui-Gon Jinn is the most powerful Jedi in the galaxy is because he had seen everything that would unfold throughout the Skywalker Saga in a vision of the future, sacrificing himself to ensure the fulfilling of the Chosen One prophecy.