Josh, for testing you can also remove the battery and connect a 6V DC power supply to the computer. Under the battery cover there is also a picofuse which may be blown if someone was tinkering with the machine before.
The HP 110 is interesting if you want to play with HP-IL - it has some BIOS routines to control the HP-IL. The other remarkable thing is that he has MS-DOS in ROM and all data on RAM disks m(which is rather limited, though). Ideally you would also want a 9114 3-1/2" floppy disk drive for it. If the computer works, you can rebuild the battery pack. The lead cells are still available e.g. under the name Hawker Enersys Cyclone 2V/2.5Ah. You can use 3 single cells (about 7$ each) and make them into a pack. Good luck Martin