The first thing you want to do is start simple. Try running it without the battery on the power brick only to rule out the battery. This problem is often caused by a battery with issues (typically cells) the BMS has not detected (and it tries to use a defective pack) or the battery was disabled but it cripples the operation of the laptop in the permanently damaged/disabled state. If it’s consistent with let’s say 5 power cycles, the battery is bad.
If that doesn’t work, the next thing to try is a replacement power adapter. A lot of the time the older Dell 130W adapters had a tendency to have sense IC problems before they fixed the problem once and for all with the 0VJCH5 and newer adapters. When these adapters fail, the laptops don’t register the adapter properly because it can’t figure out what kind it is. When the laptop can’t ID the charger, the laptop will not charge the battery or throttle it. See the next paragraph for a temporary workaround.
If neither the battery or adapter and the temporary workaround do not help, it is likely a motherboard issue. It is worth trying to remove the RAM, hard drive(s) and WiFi card to see if any of these are at fault. The problem with thse Haswell Dell laptops in general is they all tend to have a higher motherboard failure rate then early models like the Sandy and Ivy ones. It isn’t egregiously bad, but it’s common enough you need to check.
Bad charger workaround
However, it is possible to get them working again… temporarily. Once the sense IC is intermittent the adapter can not be trusted again as it will fail permanently without warning at ANY TIME. Even if it works for power, the bad sense IC may cause a problem where it doesn’t read right even if it’s fried in the sense charging will not happen but it works for power. Even if it works and this doesn’t happen, you should replace the adapter if it’s an early one because it WILL eventually fail due to this problem. I only gave them 2 chances before getting them replaced.
In order to bring the adapter back to sanity, let it sit unplugged from the laptop and wall for 5-10 minutes (or more) and/or wait for the blue LED to go out. Once this is done, it is sufficiently drained and the sense IC will behave again. Do this until the new one comes, but you can use the old one as a backup until it fails if you like and aren’t obligated to send it back. I did this on my E6540 and it worked to get me out of trouble when the adapter failed this way. Since I only needed it to live long enough for Dell to send the new adapter (~3 days), it was an acceptable alternative to being stuck. They swapped them with no fuss, so I think it’s a known fault with some of the early adapters. I have no idea how long the reset will work before it’s dead beyond resetting it this way since I couldn’t test a dead one knowing I had a good one on hand ready to go.