Self-closing hinges stopped working

The back door of our house has what I believe are called "self-closing" hinges and until recently it would close on its own when not held open. Each of the two self-closing hinges (the top and bottom ones) has a hole on its side and each hole had a pin (really just a nail) in it.

Not long ago, though, the nail / pin in one of them snapped off and the door stopped closing on its own. I took the nail out of the other one to see if there was anything "special" about it but as best I can tell it really was just an ordinary nail. Unfortunately, I'm now not able to replace the nails / pins in either hinge's hole; whenever I try something prevents me from inserting a new nail. If that were happening with just the one that snapped off I'd assume that it's the broken piece of nail still in there that prevented me from inserting a new one, but I also can't re-insert a nail in the other hinge from which I removed the fully intact nail.

So my question is this: does anyone know what I need to do to be able to get nails into the holes so that the door will close on its own again?

P.S. I don't know if this is relevant or not but the door is apparently also not very well balanced because not only doesn't it close on its own now but it actually swings to fully open when not closed.

Try removing the door with the hinges attached and then see if you can get the nail back in them. If that doesn't work, you may have to go as far as removing the hinges from the door jamb and the door to get the nail back in.

Sounds like a nail was used in place of the original set screw, which holds the pre-loaded spring in place. Find the Allen wrench fitting in the top or bottom of the self closing hinge. Using an Allen wrench and a nail (6P or 8P?) that will fit in the hole, turn the Allen wrench toward the door, using the nail in the side hole to allow you to hold the spring load and to reposition the Allen wrench for another turn, using the nail to hold the spring in place. As the Allen wrench gets harder to turn, leave the nail in the side hole, and test the self closing capability. Repeat until satisfied with its operation. You probably need to force a short screw in the side hole, which will permanently hold the pre-loaded spring in place

When the hinge is in the position it has when the door is fully closed the hinge spring is possibly totally untorqued and so you may be able to reinsert the pin.

Your door swings open because the frame is not plumb.