Lots of factors can affect wifi performance. You should mainly look at signal strength/quality, and the traffic load on the access point.
In terms of signal strength... is it bad right now? If it is, then either your location(s) are in areas of bad reception, or the wifi adaptor you use isn't good enough (unlikely).
Also, I see there's an updated driver available for it. If you have the old version, you should update:
http://support.asus.com/download.aspx?SLanguage=en&p=11&m=PCE-N15&hashedid=CqoqKk8GIW3PRlJJ
As for traffic load, you gotta remember that wifi sucks when it comes to sharing bandwidth between multiple people. The more people are there using it at the same time in a given area where the same AP gives coverage, the higher the performance penalty per person. It's a lot worse than sharing a wired LAN. Maybe the ones who can play LoL are in a different area, served by a different AP?
If it's not that, then it could be an incompatibility between your adapter's chipset and the one in the school's AP, but this is unlikely as well.
Anyway, what I'd do is update the driver, and play around with the antenna angles on your wifi card. You gotta remember that antennae like the ones on the card radiate in a "bagel" shape around it, so you want the antenna to be perpendicular to where the AP is. By contrast, pointing an antenna at the AP will give you the worst signal.
I wouldn't buy the USB thing you linked, because: 1) it's a miniature design that doesn't have external antennae, so its signal will be weaker, and 2) it's also an ASUS adapter, so maybe they use the same wireless chipsets.. which would be bad if the chipset incompatibility thing is the cause here. If you can't resolve the performance issue, yet your room mate or neighbour who's on the same AP is having no issues, look for another wireless card from another brand. Or try getting a stand-alone external antenna.. like
this one, idk.