Jaime Lannister killed Aerys II Targaryen before the events of Game of Thrones, so how old was the Mad King when he died? Although Game of Thrones takes place several years after Robert's Rebellion, the effects of that moment in Westeros history are still keenly felt. The years turned Robert himself from a great warrior to a drunken king, the remaining Targaryens had to live in secret exile, and Jaime could not hope to escape the Kingslayer moniker.
Despite being a member of Aerys' Kingsguard, Jaime decided to slay the Mad King when the time came. Although the act he performed is later revealed to be for the greater good, because the Mad King wanted to burn the entirety of King's Landing to the ground, the fact he turned against the ruler he was sworn to serve followed him for the rest of his life. As for Aerys himself, it was the end of a turbulent reign that had actually once started out with promise, before descending into paranoia, villainy, and of course, madness.
The person who would become known to history as the Mad King, Aerys Targaryen, was born in 244 AC (After Conquest), some 54 years before the events of Game of Thrones. As confirmed in works such as Fire & Blood, Aerys was crowned in 262AC at just 18-years-old, although his coronation took place in 259AC in Game of Thrones' timeline, which shifts things forward slightly to age up its characters. In the books, the Aerys was killed at the age of 39, in the year 283AC. In Game of Thrones, the Mad King was around 37-years-old when Jaime killed him, since this event took place in 281AC (it's unclear, though, if his birth year is the same, since the show changed Aerys' father from Jaehaerys II to Aegon V, so it's possible he is older).
Similarly, Jaime's age also shifts between Game of Thrones and A Song of Ice and Fire. In the latter, he was born in 266AC, making him around 17 when he kills Aerys. In the show, his birth year is 262AC, meaning Jaime was around 19-years-old when he killed the Mad King and forever became known as the Kingslayer. In both versions, though, he was the youngest person to be appointed to the Kingsguard. Both of their ages are quite striking in how relatively young they are, regardless of whether looking at Game of Thrones or A Song of Ice and Fire.
For the Mad King in particular, his life was remarkably short given just how much happened, and how much he transformed over the course of it. The young Aerys was known for being a happy and charming boy, if an ambitious one, and that was something that even continued into the early part of his reign. Over the years, however, he became increasingly paranoid, and especially distrusting of his Hand, Tywin Lannister. Although he was declining before then, the major turning point for the Mad King going crazy was the Defiance of Duskendale, which took place when he was around 33-years-old.
After being held prisoner for six months, he was plunged much deeper into madness, leading to the realm needing to rally against him, and his life being cut extremely short. Interestingly, the Mad King is generally depicted as looking far older than his years - David Rintoul, who played him in Game of Thrones, was well into his 60s when he took on the role. Since Aerys didn't let people cut his hair or nails, wouldn't wash, and ate little as he feared being poisoned, he became thin, gaunt, and withered far more than his years would suggest.