Try with battery out and mains adaptor only, even if you think battery is fine.
It could be almost any problem. May take a while to narrow it down.
To check for dodgy jack, wiggle up/down/left/right while trying to turn on.
Your data is safe enough. You can remove the hard drive and plug it into a PC with the correct (probably SATA data and power) cables.
Or out it into a USB caddy/enclosure to connect to another machine.