Is my switch completely broken?

Okay, when trying to diagnose charging issues, it's always best to start at the source and work your way into the Switch. So the place to start is with your charger; you'll need to verify that the charger itself is working, either by swapping it out with a different one, or trying the charger with a different device. Once you know you're getting power from the charger, the cable is next; again, either try a different cable or try the cable with a different device.

Now that we've eliminated all of the external factors, we want to take a look at the charging port itself. Take a magnifying glass and a bright light and check the port for any dust, dirt, or any other kind of foreign object inside the port. Clean the port with some 90% or higher isopropyl alcohol, maybe a microfiber eyeglass cloth and a toothpick.

Assuming it's still not charging, try wiggling the jack when it's plugged in; push to either side and up and down and see if you get any indication of charging in any position. If something connects with pressure on the plug, then you've likely got a damaged charging port that will need to be replaced. Unfortunately, it's a difficult job as it's soldered to the motherboard and requires a hot air station in order to change; it can't be done with just a soldering iron, so most of us would have to take it to a repair shop to have that done.

After that is the battery. Sounds like you have that covered as I assume you've ordered a battery, so if nothing has helped up to now, then that's the next step to put in a new battery.

Finally, if the new battery doesn't fix the problem then the only thing left is a problem on the logic board itself that requires board level repairs by an expert.

So check out the things I've mentioned, and if need be swap out the battery then let us know what happens. Good luck with your repair!