XPS-15 Will not turn on

Check if the light is lit at the plug end of the AC adapter or on the adapter block. And see if that light remains lit when you plug it into the laptop's jack. You said that a power plug light is on, but I don't know if that is on the adapter/plug, or on the laptop. If you have or can borrow another Dell AC adapter of at least 130 watts, try it.

It could be that the charging jack needs to be replaced. First, I would try some other steps.

1. Disconnect all peripherals: printers; USB devices; media cards; USB hubs and extenders; etc.
2. Disconnect the AC adapter.
3. Disconnect the battery.
4. Press and hold the power button for 30 seconds to discharge any residual electricity.
5. Reconnect the AC adapter
6. Try to boot the laptop.

Examine the main battery. That laptop is ten years old. If that is the original battery, it's at or near death. Even if you replaced it once, the replacement may now be pretty much dead. Worse, it may be swelling, which can damage the laptop. If there is any sign of swelling,, you must remove it and safely dispose of it.
Swollen Battery Information and Guidance | Dell Canada

If the list of steps doesn't allow booting does nothing, then I would repeat steps 1-4, and then remove the CMOS battery (coin-cell battery). After 30 minutes, replace it with a brand new coin-cell battery. Then connect the AC adapter and try to boot.

Follow the disassembly and reassembly steps in Dell's Owner's Manual.
Look on YouTube for videos showing disassembly of your model. But, be cautious with them, as some use steps or methods that are not appropriate. Use the videos to add to your understanding of what to do, but follow the steps in Dell's manual.

Whenever touching components or working inside a computer, wear a grounded wrist strap, also called anti-static wrist strap, ESD wrist strap, or ground bracelet. It's a cheap and sensible precaution. Rest the laptop on an anti-static mat or at least a reasonable alternative such as corrugated cardboard.

EDIT: I see ejn63 posted as I was typing. dclark15773, you've now got some work to do.

Verify that the LED on the AC adapter comes on when just the AC side is plugged in.  If it doesn't, replace the adapter.

Unplug the system, remove the base cover and disconnect the main battery from the system board.  Hold the power button for 30 seconds.  Remove and reinstall the memory modules.  

Connect just the AC adapter and see if the system shows any signs of life.  If it does powerup, shut down and reconnect the battery and try again.  If that halts powerup, the battery is bad.  

If the system still won't power up, the mainboard is bad.

The system is old enough that mainboards are at or below the $100 mark.  The downside is that they'll be of similar vintage to the system, with an unknown amount of life remaining.  And even before any labor charge that's about 50% of the total value of the system, so it may make little sense to replace the board.