There might be a short circuit inside the camera that shorts the battery leads - perhaps intermittently - so they discharge quickly. Or the voltage regulator may be broken (if those exist in cameras, I'm not sure). Third option is the batteries are old, have been subjected to heat etc. I would try another battery to be sure. Old or overheated batteries tend to last shorter.

You could try to drill a tiny hole in the jack from the outside, then screw a wood screw into it, making a handle with which to pull it out.