This is scarier than at first thought. You are going to need to remove the 5300 board and inspect the underside of the powersupply/recharging circuit area. Check for the obvious - corrosion in the area which might be shorting out the circuit. Clean out and resolder any bad solder joints you find.
Also check the insulating plastic on the 5300 case. Make sure there are no holes where the 5300 board could short out against the RF shield of the 5300 case. The clips to the battery from the board also needs to be looked at and resoldered if necessary.
I will dare say that there might be a bad regulator on it but that will be difficult to fix. If the 5300 turns on with the power pack alone, then it should be enough to do other things you need to do with it.
BUT - I know this from a ThinkPad 560E that I have. It has a 1/4 shorted battery, which wont charge and wont power the unit, and the 560E runs hot with it plugged in. Given, it is partly shorted out. But when the hard drive failed, I replaced it with a CF/IDE SSD in it. Now, believe it or not, the battery charges 3/4 of the way but never full as that part is shorted out and it runs the machine for about an hour without being plugged in! I'm thinking that the battery shorted out when the hard drive was going bad years ago, and with the hard drive bad, the battery was never powerful enough to run it and the laptop too. When plugged in (when the bad hard drive worked before it died) both it and the battery were getting hot because of the high current draw the hard drive was pulling.
But does this apply to the 5300? Only way to test it is to do several tests - run it without the battery, run it without the hard drive and run it without the floppy drive. Something in the system is pulling to much power from the sounds of it. Thing is what? If the power supply area is bad, only a couple of people here might be able to fix it. I'm not one of them as I do not have the tools to fix it with.