Buddy gave laptop for debt and now it won't turn on. Trying not to KHA

My buddy owed me some money so he gave me an alienware 17 r1 P18E001 as payment. It didn't have the power cord so I purchased a real Dell, not an aftermarket, power supply. I hooked it up and got it to start up but there was an issue. Once I shut it off or it lost power and shut down, whether done properly by me or was disconnected or died, it wouldn't turn back on right away. I would push the button and get nothing. Sometimes the light around the power would be on and other times it wouldn't. I would try all the tricks. Disconnect battery, hold button down everything and nothing. I would let it set and come back to it hours and hours later and it would start right up, plugged in or not, battery attached or not, light on around power button or not. After a long time it would power right up. But if it was shut down or anything then I would have to wait for a long time for it to start. Well, now it won't start at all. Power is plugged in. Sometimes the light around the button is on. Mainly no light around the power button but I get nothing when I push the power button. Power supply is new. Battery did hold a little bit of a charge. Not sure how long a battery on that laptop should last but I could get a couple hours. Think it might be the DC power port or maybe the power button or both. Just need some feed back and need to know how to troubleshoot to find issue. Thank you.