Timeline of Daenerys’ and Jon’ births
I’ve seen some people confused about Jon’ and Daenerys’ age, so I thought I would write an explanation.
Let’s start with Aerys and Rhaella, the parents of Rhaegar and Daenerys. They married at a very young age (Aerys was 15 and Rhaella either 13 or 14). Rhaella gave birth to Rhaegar in 259 AC, at the age of either 14 or 15. Viserys was born in 276 AC, so 17 years after Rhaegar (Rhaella had many miscarriages, difficult pregnancies and stillbirths during these 17 years).
In 282 AC Lyanna and Rhaegar disappear together - in 283 AC, the Battle of the Trident takes place and Rhaegar is killed. Few days prior to the Battle of the Trident, Aerys rapes his wife Rhaella, and that is the night when Daenerys is conceived.
Relations between Aerys and his queen had been strained during the last years of his reign. They slept apart and did their best to avoid each other during the waking hours. But whenever Aerys gave a man to the flames, Queen Rhaella would have a visitor in the night. The day he burned his mace-and-dagger Hand, Jaime and Jon Darry had stood at guard outside her bedchamber whilst the king took his pleasure.
The “mace-and-dagger” Hand is Qarlton Chelsted. Aerys burned him alive after that one discovered the wildfire plot and tried to stop it.
“Aerys’s new mace-and-dagger Hand was not utterly stupid, and with Rossart, Belis, and Garigus coming and going night and day, he became suspicious. Chelsted, that was his name, Lord Chelsted.“ It had come back to him suddenly, with the telling. “I’d thought the man craven, but the day he confronted Aerys he found some courage somewhere. He did all he could to dissuade him. He reasoned, he jested, he threatened, and finally he begged. When that failed he took off his chain of office and flung it down on the floor. Aerys burnt him alive for that, and hung his chain about the neck of Rossart, his favorite pyromancer.”
When Aerys hears that Rhaegar lost, he sends Rhaella and Viserys to Dragonstone.
"Rhaegar met Robert on the Trident, and you know what happened there. When the word reached court, Aerys packed the queen off to Dragonstone with Prince Viserys. Princess Elia would have gone as well, but he forbade it. Somehow he had gotten it in his head that Prince Lewyn must have betrayed Rhaegar on the Trident, but he thought he could keep Dorne loyal so long as he kept Elia and Aegon by his side.”
The sack of King’s Landing happens a fortnight after the Battle of the Trident, Eddard arrives to King’s Landing shortly after the Sack, and few days later he leaves for Dorne to look for Lyanna. He soon finds her, and promises to protect Jon. Jon was born in 283 AC.
Meanwhile Rhaella stays on Dragonstone (Robert’s forces didn’t try to take the castle right away) until her pregnancy reaches its term. She delivers Daenerys and dies in childbirth. Daenerys was born in 284 AC.
Jon is older than Daenerys by around 8 months.