While we often point people to count the cycles as most folks run their battery hard, its not the only measure, Age is also a factor! Given its age I'm sure this is the original battery! Batteries only last for three to four years as the chemistry breaks down.
The left side is where the charger connects the MagSafe connector can often heat up (warm) as the power feeding though it has enough resistance, but a weak battery also demands more power pushes things getting it very warm!
So to fix things you need to make sure the MagSafe connector interface is clean as a dirty connector can make things worse. You may need to replace it if it's become warped. MacBook Air 13" (Mid 2013-2017) I/O Board and he's the guide MacBook Air 13" Early 2015 I/O Board Replacement
But even that is not enough! You need to also replace the battery too! MacBook Air 13" (Late 2010-2017) Battery