How does one recover from a connector suddenly disconnecting?

tl;dr what do I do with a broken connector? Can I fix this?

CONTEXT:
My 3DS (New) has been out of commission for 2 months, so I've been troubleshooting here and there. I knew top screen and speaker were damaged, but I didn't rule out the camera. I replaced them at different points last month. At some point the parts connecting this connector near the bottom screen ribbon connector came loose, hanging by a thread. Still, after replacing the camera, I made some progress, or so I thought.

Today, I was replacing the speaker, but as soon as touched this connector it suddenly disconnected from the board…I was shocked, but perhaps earlier mistakes from my first attempt in January came back to bite me.
My screen did show life when the SD card was in, but without it I get the same pause and pop sound (before replacing the canerea I didn't even have that – blue LED instantly shut off after about 1 second without SD, 10~20 frames with SD, and without SD it faded out while with SD there was no fading). At least I know the screen works now. But I think the broken connector is preventing it from passing the checks to power on.

Anyway, what I want to know is: How do I get around this issue? Solder(i saw a reply on an older post saying it's not a good idea but idk)? Paper to put pressure down? Maybe some combination to keep broken pin on the proper place? I'm a bit lost