Right.. Well there's only a few ways to test this, but both involve a kind friend or money lol.
The free way is to borrow someone elses power supply. As long as its 19v and about 3a it should be fine.
If it charges off that, its your psu, if it doesn't, its possibly battery.
The other method is to take the battery out and get a vault meter on the terminals on the laptop. 2 of these in some combination should kick out 19v DC. If it doesnt then an internal fault could be to blame.
The 3rd is to ignore the battery health meter (as at the end of the day as long as it has a charge its going to say its fine) and buy a new battery.