Need help fixing 3DS that only powers on when plugged in.

Hi,

I have an original 3DS that only powers on when plugged in to the charger. (It charges fine.) However, I can disconnect it from the charger immediately after it boots up and it works fine with just the battery. I have changed the battery and it turned on without the charger a couple of times then went back to only turning on when plugged in. I have tested the batteries with other 3DSs and they work great. I have taken the 3DS completely apart and cleaned it. I did notice a little rust underneath the SD reader. I cleaned it up really good and the SD reader reads the SD just fine. I even tried removing the SD card and putting it back in but this does not work. I tried without the SD card, this does not help. I also removed the battery and then put it back in, did not work.

I have fixed, repaired, and taken apart multiple 3DSs and DSs. I have even done some basic (I mean basic) soldering. Cannot seem to figure this one out. Help please!