Since you don't have the ability to perform a full reset (well, you can, but you won't have any personal data on it anymore, including whatever videos you're watching), I have a couple of suggestions.
First is a strange bug that I know has affected some iPad users and so might be an issue for you; it's one of those "how can these two things possibly be related" bugs.
Launch the Settings app and choose General=>Reset=>Reset Network Settings. All this will really mean is you'll lose your saved WiFi networks. If that doesn't solve it then I'd try the next suggestion (below), and if it doesn't work then in this same area choose Reset All Settings; I suggest trying this one third just because it means all of your apps will lose their stored settings and revert to their defaults, which is a hassle to manually restore.
Second, is it possible that your iPad is full or near-full? I've heard of this problem when there's so little space left that the Videos app can't even clear out any caches and so is stuck.
If so then you might try deleting a few things that you can get to. For example, delete some files in the iPod app, or delete any particularly large apps that you can live without until you next synch. Large apps include complex games (pretty much anything that renders 3 dimensions like first-person shooter, racing games, etc.) and apps with a lot of documents like iBooks or Pages or what have you. You can also delete photos via the Photos app if you have a bunch of large ones that are stored elsewhere.
After clearing out some space, restart your iPad again.
If all three of those fail then you're pretty much stuck with resetting your iPad, but again, that wipes everything and returns the device to its factory settings, which is likely not ideal in your situations since a restore requires your computer.