In other words, you won't be able to get rid of the viruses on the computer without being able to install something on them.
So you're saying that this computer center is being shut down anyway in a few months, and it sounds like the people in charge of them can't be bothered to fix them or give you control. It sounds like you have nothing to lose, so I'm going to recommend a radical solution. Other than continuing to cajole your sysadmins or admitting defeat, the best option I can think of is to to back up any files that you might need from these machines and then wipe their hard drives and reinstall a new operating system. Essentially, just take them over from the deadbeats who are supposed to be taking care of them. It's what I would do. Then, don't install anything on them except a new web browser (Firefox or Google Chrome) and use Google Apps to do word processing and stuff. If you use a safer web browser and don't install anything on them, and especially if you don't use Windows (read on) then you're less likely to get viruses or malware in the future.
If you wanted to reinstall Windows, and you don't have the install disks, you can see if you have a sticker on the side of the computer somewhere that has a license key number. For example, if they're Dells, there's probably a sticker. It will tell you what version of Windows it's licensed for, and if you can find a disk of that exact type, it should work. If there's no sticker, there are ways of recovering the license key before you wipe the hard drive.
If you can't find disks anywhere, you can find places to download Windows XP, though it's a violation of US copyright law. Of course, since those computers have legal Windows licenses and Microsoft has already been paid, you may consider yourself morally entitled to download Windows, if not legally entitled. If you're going to go that route, you shouldn't bother with the standard Windows disks. Do a google search for TinyXP. It's a specially-customized version of Windows that some people consider to be the best version of Windows available. Of course, going this route involves being able to find and download illegal software, and burn it to disk. If the PCs in your lab won't allow you to download the the software you'd need to use to do this, or don't have CD burners, then you'll have to use another computer to make the disks. I don't know what your level of technical proficiency is, but you might find it daunting.
But I'd serously consider installing Linux. Download Ubuntu. It's free, easy to install and use, and as long as there isn't any particular piece of Windows software you'd need, it will work fine for helping kids learn about using the internet and doing basic personal computing. It's 100% Legal to download Linux, and Ubuntu will even send you a free install CD (though there can be a long wait) if you can't burn one yourself. Unlike Windows, it's vurtually impossible for you to get a virus on Linux. You don't need any anti-virus software, and an added bonus will be the the kids in your computer center will be less likely to mess these machines up by downloading crap from the internet or falling prey to scams that lure them into installing spyware or other harmful applications. All of that stuff only works on Windows. Another advantage is that almost all consumer Linux software is free to download, so you'll have a huge library of interesting programs to choose from, including games, if you'd like to go that route.
Any of these options will take a couple of hours of prep and install time, but once you've done the first computer, the other four will go quickly. If your shiftless sysadmins ever showed up, it's likely that they would just reinstall Windows themselves, if these computers are as messed up as you say they are, which wouldn't be any faster than what you need to do. If they ever do show up, and see what you've done, my guess is that they'd be a combination of insulted and impressed.
If I were you, I'd go the Linux route, and try to include the kids in the process of downloading, burning, and installing the OS. Have them do a little background research into the difference between Windows and Linux, and where each came from. They might be interested to learn that Linus Torvalds, the guy who started Linux, was a college student in Finland who didn't work for a big company and didn't have any particular resources other than a good education, and now millions of people, including huge companies, use his software. It's also worth pointing out that it's due to the contributions of tens of thousands of individual volunteer contributors that modern-day Linux exists, and that it doesn't take anything more than a little know-how and a desire to make things better to be a part of a huge and important movement like Linux. I think you can take those infected, broken down computers, and turn them into a great experience for the kids you're working with. Good luck!
I'm sure other OSNews readers will have some good advice in the comments below, or will contradict me, or both.