Minecraft has to be installed on the computer to work, but if you're up for a bit of work then there's a way around it.
If you install Linux on your USB memory stick then you'll be able to boot the laptop directly from that, bypassing Windows altogether. You'll need to set it up once you've got it running so that it works properly on her machine, in particular you'll need the graphics driver installed. Having done that all you need to do now is to install Minecraft in your Linux OS. Now you've got a USB memory stick containing both your operating system and game which you can boot any computer from and then carry on with your game. Try to get hold of a fast USB memory stick because it's got to do the job that the hard drive normally does.
Of course though before you do any of this you should check that her laptop can be booted from a USB memory stick. As it's a school machine that ability may have been locked down for security reasons and impossible to change without knowing the bios password
Another way would be to just copy your worlds to your USB memory stick. Then when you want to play download and install Minecraft because it only takes a minute, then copy your worlds onto her laptop. They're kept in the following folder:
C:\Users\[user account name]\Appdata\Roaming\.Minecraft\saves
The Appdata folder is normally hidden so open the Windows file explorer and type %appdata% in the address box to navigate straight to C:\Users\[user account name]\Appdata\Roaming
After you finish playing go back to that folder and copy your saves folder back to your USB memory stick before you log out to keep what you've been doing in Minecraft, ready to copy it back in again next time.
Linux is free
Edit: I'm always happy to help, and it's nice to find an interesting question. I think you'll find the second method, just copying the saves folder in and saving it back to the memory stick afterwards to be much easier. When you've done it a couple of times you'll find it easy and it'll only take you a minute.
