Hi, I thought I would give my two cents.
A few things: what you are describing as the wifi-module (the removable socket thing connected to the black wire) needs to be inserted while the DS turns on otherwise you won't power to the displays.
The symptoms you are describing, turning on green, flickers orange, turns off, and the square with 4 corners EM10 are related.
From the picture, it looks like your EM10 and/or your F1 are damaged and likely the reason for your problem.
You can test which is the issue
1. By plugging in the charger and seeing if the orange light stays on - if it does, this means the circuit to the battery is complete (F1 is fine) and is charging the battery (charge port fine).
2. To test if the battery is working, make sure you use a battery you know works, charges, and can power on a DS. Remove the charge cable, and attempt to power on the console with this working battery. If the device powers on, F2 is fine and the DS can boot on battery (F2 fuse is located by the bottom right corner of the DS cartridge reader). Depending on your board, F2 may have 32 on the top, F, or is a solid green block. Don't worry, these are all variations of fuses.
3. If the DS does not boot on battery, and ONLY will power on if it's connected while charging, then F2 is bad.
4. If you get the device to power on, BUT
ONE of the screen flashes then console shuts off, the issue may be the screen (will stay black while other flashes white). If this happens, you will need to test with a
different working screen (in place of the dark screen) to make sure it was just the screen and not another component in the circuit that is broken.
5. If you get the device to power on, BUT
BOTH screens flashes white and shuts off, then you have an issue with the battery. You will need to replace the original DS battery with a new or different working one.
F1 is the fuse for the charging ciruit portion of the DS, and F2 is the fuse for the battery powered portion.
I could keep going but I would suggest first replacing the F1 and EM10. This will require a solder iron. If you don't have a solder iron, you are not going to be able to fix the broken DS.
Hope this helps.