Something was already horribly wrong. At that point a good
test might've been memtest86+, to visually check dust levels
and fan operation, to measure voltages, run a hard drive
manufacturer's diagnostic, and try different drive cables.
I would first suspect a bad drive or bad drive data cable.
Win95 is faster than 98 but no it is not the solution as the
added *bloat* (features) of Win98 are not the cause of the
problem. A Pentium2 system with 192MB can even run Win2k,
barely WinXP ok for basic tasks. It is a hardware issue and
if Win98's features and (in general, ignoring the problems
on this system) lesser stability than Win2k or XP are
acceptible, then Win98 is a good choice for that system.
No, in '95 Win98 hadn't been released yet, only a very odd
combination of very old parts would have been used at the
point when Win98 was released, because by that point even
the low-end cheap stuff used a K6-2or -3 if budget was the
factor.
On the other hand, a P120 can run win98 better than OP's
grandmother's system was doing so, providing it had enough
memory (at least 32MB was a ballpark figure with the next
32MB+ added being a significant performance increase).
Delete the unneeded files to free up space and then see if
it still is slow.
Since there was a problem right at the start installing
win98, taking so long, investigate that first. If it
weren't for that I would've suspected a virus or malware,
though these could also be present.
It depends on when it's slow, a bad cable can interfere with
I/O so much that it is perpetually sluggish every time I/O
is attempted. Also check Device Manager to see if the drive
is operating in DMA mode as it should be, or PIO mode which
it should not (unless the drive were PIO-mode capable only,
which would make the drive significantly older than the
system itself as all P2-300 era drives were DMA capable).
Lastly, if the system is going to continue to be used, check
on a bios update that would support larger drives, but don't
flash it yet until you know the system is stable (I suggest
running Memtest86+ for several hours to see if there are
errors or crashes first). Also check that the floppy drive
reads well as you don't want a read error during flashing.
After you have a bios update (or not and you either put a
PCI IDE controller card in to increase HDD capacity support
or just use the largest drive supported even if it means
using a drive capacity limiting jumper), consider replacing
the hard drive because new drives are so much faster, and
that drive is old enough it could fail at any time (and
might already be failing).
There is no reason to install 3.1 or 3.11 on a P2 w/192MB
memory just to solve an OS overhead issue. Win98 runs very
well on a P2 w/192MB memory until it comes time to do
something really demanding outside of basic GUI functions or
office, like video compression, gaming, etc.
It doesn't sound like the operating system. Did you start with a format
of the hard drive. If not you should have because there could be all
sorts of things in files that could slow it down. Maybe it would be
worth it to get a new hard drive. Since it is for word processing it
could be the cheapest one available. There is nothing that should have
slowed a computer down that much. If you formatted a new hard drive and
put Windows 98 on it there should be no problems with it running that
slow. It just doesn't make any sense.
Lots of things could have gone wrong on a system that old.
It might have bad caps, residue on connectors, bad drive
cables, a PSU barely working, etc, etc.
At some point one has to accept that nothing lasts forever
and the system was reasonably good to last the 10 years or
so it did. It might be time to just replace it with a
something modern... to get a long forward-looking lifespan
if nothing else, it need not be an expensive system, I
suggest looking at something from Dell or HP that can be
bought with WinXP instead of Vista.