Sorry about the exceptionally long delay. I kept reminding my sister, but she kept putting it off as she didn't have the time. About a week and a half ago her pre-installed Norton antivirus expired, and she had trouble installing Avira AntiVir as well as AVG. Something about a redistributable kit that she didn't take care of. Not long after, she got a virus, and it wouldn't let any programs other than Internet Explorer and Firefox run, including Java (so I couldn't use an online scanner like Kapersky). She traded her laptop for another one we had, so I ended up taking the hard drive out, putting it in an external enclosure I had, and removing the virus through MalwareBytes Anti-Malware on another computer. I attempted to go to the recovery console and do any chkdsk (including with the "/r" argument), and although it got farther in the second step than before, it still got stuck.
I ended up backing up every non-system file she had to an external hard drive (plus a few folders in "program files"), getting serial numbers for her programs with a couple of free utilities I had, and used the recovery console to restore the computer to factory default. After it was restored to factory default, I downloaded Avira AntiVir, Firefox, and FoxIt PDF Viewer; installed them; punched in the serial numbers to any programs that needed it; restored her files; and updated the computer fully with Windows Update. I then gave her the two text files containing the serial numbers, and gave the computer back.
Now Windows Update works again, her webcam functions normally, and every other problem she had was solved. Although it was probably not necessary to go to that extent to fix the computer, it would've taken an incredibly long time to correct the problems listed in this thread, and
the newer problem of programs not wanting to install, using any other method. I thank you for all your help, and I did use the recovery disk you linked me to, but unfortunately I can't give a solution for those experiencing the same problem in this thread. But the computer is fixed, so you can mark this thread as solved.