[Al Skierkiewicz]

Indie,
First, NiCd batteries are 1.2 volts per cell so a three cell should be listed as 3.6 volts. Normal for an appliance of this size is two cells which is the 2.4 volts you list.
Sealed appliances are generally glued or ultrasonically welded. While yours is over ten years old, I doubt that the seals on the moving parts have held up this long. That being said, it is likely that there has been some water entering the device over time.
A Dremmel tool can be used but that removes material that you will not get back so the parts won't easily seal after a repair. A technique I have used from time to time is chucking the device in a vise and squeezing until part of the seam opens. This takes a lot of patience so that you don't over do the squeeze and compress the internals.
NiCd batteries have a finite life much like our robot batteries. Often though, the battery grows a "fuzz" around the terminals that eventually shorts the battery cell. The other life limit is caused by out-gassing of the chemicals. This is a naturally occurring event that is a result of charging. Once you have the device open, you need to measure the cell length and diameter and then purchase replacement cells from DigiKey or other outlets. Generally, these size devices will use sub-C cells but not always. I would buy cells that have solder tabs attached so that you can make the repair. Batteries with no tabs have a stainless terminal that requires special flux to solder.