I did a system restore, which I thought would bring it back to the previous good point. Instead, it completely reset to factory defaults.
Microsoft's System Restore and Dell's recovery to factory settings (aka, factory restore) are two different things. If it reverted the computer to the way it was when it came from the factory, then you did a Dell restore.
I wasn't sure whether that would format the disc or not. So is that factory reset the same thing as a full reinstall with formatting the disc?
Yes, the Dell factory recovery is the same thing as reformatting and reinstalling from a Dell reinstallation disc.
You can tell whether that's what you've done by the time it took and the result. A factory restore should take about 10-20 minutes. The result would be you'd have only factory-installed programs (any user-installed programs and user data would be gone) and Windows Update would be way outdated. (The Windows Update status would revert to the date the computer was shipped.)
It was working fine after that, then I took it to my sister's house and it was slow again, nothing else running, connecting to a hot spot. It took 3 or 4 minutes to bring up a browser, over 5 minutes to shut down. I brought it back to my house, and it did the same thing.
And I gather you've ruled out slow internet? Was it slow bringing up random programs, or only the internet browser?
Another possibility to consider is that if you did do a factory restore, Windows Update would be so outdated it could have been tying up the computer for quite some time as it would have been working to bring itself back up to date.
...but maybe you're right about the problem being the hard drive.
Find out the brand of hard drive and download a test/diagnostic utility from the manufacturer's website. It wouldn't be unusual for the hard drive to be on its way out.
I'll look for a recovery disc option. I don't know if she ever made discs.
Most Win7-era Dells often shipped with "Dell DataSafe Local Backup", and the option to create recovery discs would have been in there.
If the computer isn't too old, there might also be an outside chance you could
download a recovery image from Dell. It's worth a try, but to be honest, 90% of the time I've tried that site it came up empty.