They're cheaply made and they die all the time, just look on the Amazon reviews. With all due respect they're $20 not $100 and you get what you pay for. You'll probably have to buy another one, and get the warranty next time. I'm only assuming you didn't or else you would be able to just get another one for free.
If it was just the LED it should still sync your controller but since it's not I don't see how it's the LED.
I actually just bought one myself for both my Switch and Wii U, but I'm not going to be shocked if it dies on me also.
If it's working only when plugged in there's something wrong with the wireless part of it.
There's not much troubleshooting in it it either works or it doesn't for the most part and it's obviously not working.