If you only need to get info from the PC's Hard Drive, two options I'd try in no specific order are (A) put the Hard Drive in a working machine and copy the data across or (B) With the Battery dead, the CMOS Config Settings on the MotherBoard will have been lost. Most likely cause of failing to boot is that CMOS shows no Hard Drive installed, or anything else. i.e. A Drive, etc... Look for a label on the HardDrive showing Cylinders, Heads & Sectors info. Turn on your PC, hopefully you'll see something like "Press to enter setup" (or some other key combination) Enter setup, and check that there is a Hard Drive INSTALLED, and enter correct C,H,S info and check that the it can be detected. If possible try AUTO Detect if that is an available option. If that gets it booting, all will be okay as long as you keep the PC powered on until you get your data off it.
Incidentally, from the description, I would say the battery you describe is soldered to the motherboard. As old as that board is the battery may have leaked and permanently damaged the motherboard. So your only option might be to concentrate your efforts on the hard drive itself.
Chris J, November 2006