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