*sigh* Okay... you have two separate issues here.
One, yes, your power connector is broken and should be fixed. HOWEVER, from your description, it sounds like you can push the plug in, set it just right, and the laptop will work, correct? So that means the laptop IS getting power at that time... and if the power is getting into it, then it WILL charge the battery.
Normally, if the laptop is running, the battery should be charging, and if you disconnect the power, it should still run on the battery... BUT if you bump the plug, then it loses power and shuts off... this means that while the battery may be RECEIVING power from the power jack, it is not ACCEPTING or HOLDING a charge.
So you can fix the jack (or have it fixed), and it will run normally as long as it's plugged in... but you will still need a new battery if you want to run it unplugged.
The other possibility is that your use of the wrong charger has fried the charging control, so that it's NOT sending the battery a charge... but IF that's the case, then it's probably killed the battery as well.