You will need to take it apart and see exactly what has happened to it.
If it truly has swollen or is otherwise leaking - you will want to get it out of the system!

Most laptop work fine without a battery installed - but I have run into a few over the years which relied on the battery as part of it's regulation and could damage themselves if it wasn't installed (although if yours is badly damaged/swollen it might effectively be "not installed") - some searching on the exact model of computer might turn up information about how well it can be expected to do without a battery.

"swollen" isn't a NMIH battery failure you can expect to fix!
The most common failure of NMIC and NICD batteries is internal shorts which develop during "reverse" charging - not as uncommon as you might think, these batteries are almost always a number of cells(nominally 1.2v each) connected in series to get a higher total voltage (eg: if your pack puts out 12v, it has 10 cells in series) - and since the cells are never *exactly* matched(which is impossible) one or more will almost always "run out" slightly before the others and gets "changed" a bit in reverse as the others continue to operate. (this is why you should NEVER completely discharge your pack - unfortunately these types of cells lose capacity if they are regularly NOT fully discharged, so it becomes a bit of a "game of luck" to keep them fully working)
- This type of failure can be "recovered" to some extent - the internal shorts can sometimes be blasted away with a BRIEF high current - the most common way to do this is to use a large capacitor (say 5,000uf) changed to a few volts and discharge it through the SINGLE shorted cell (in the correct polarity for changing) a few times ... but this type of failure does not cause swelling - so I'm pretty sure this is not your issue!

The good news is that NMIH cells are very common and easy to get - most likely the ones in your pack are what's known as "Sub-C" in size - a bit smaller than a "C" dry cell.
A lot of battery places can rebuild your pack for you, or sell you the cells if you want to rebuild it yourself.

Thanks for the answer, but in the meantime I noticed that my NiMH mention was simply based on that other thread and what they said. I did some further research and it looks like this Compaq was sold in both either NiMH or an 'enhanced' NiCD. So I am not really sure what battery type the unit of mine uses and it could be NiCD instead.

In my experience NiCD and MiMH are so "close" that they are often interchanged as products evolve.

NiMH is the "newer" tech, meaning products using NiCD sometimes -> NiMH but far less common the other way around.

Once you get it apart you should be able to see the part numbers of the cells and identify which type it is.

Technically there are slight differences in charging ... but unless you are going to be using it as a "daily driver" with it plugged in/charging all the time, I doubt you'd experience any real problems if you get the "wrong" type.