Chosen Solution
Need to know what to buy.
@ellenmaloney - Your jumping around a lot here and I’m not sure you’ve done anything useful with your keyboard manipulations. Take a look at this Mac startup key combinations. But I don’t think these will help you here. The D Key would have gotten you into diagnostics but if your system is not staying on this won’t help. Your symptoms sound more like either a bad battery or your logic board is shutting down from over heating. Some how we need to get your system to run longer so we can get more useful diagnostics. To test the battery and the charging logic we would use CoconutBattery. To test for a thermal issue we would use a good monitoring app like TG Pro. But neither will help us here unless we can get the system to run a bit longer. So how to get that to happen? One way if the battery is the issue is to cool the system down (battery) by placing the system into an air tight plastic bag, squeezing out all the air and putting the system into a refrigerator (not the freezer) for a few hours. In the mean time load up off a second Mac a bootable USB drive with the two apps installed. Then we just pull the system out of refrigerator leaving it in the bag for a few minutes so it not to cold the condensation forms. Then booting up under the USB drive and quickly check the battery with CoconutBattery. Take a snapshot of the main window and paste it here for us to see Adding images to an existing question. If you can lets get a snapshot of the main window of TG Pro as well (you may need to stretch the bottom a bit so all of the sensors can be seen).
If the lights come on after pressing the power button but nothing comes on the display then it would most likely need a reflow. The reflow method can be found online and is only a temporary fix for issues related to gpu’s.
Try draining the power out of the machine by pressing the power button for 40 seconds while having no battery or ac connected.
