I've checked and I can't feel any kind of blockage around the plug area, and there wouldn't have been anything on my dishes that would have caused a blockage (no leftover chunks of food for example). I live in an apartment building and the landlord is always fiddling with the plumbing one way or another which often results in him turning off the water for several minutes at a time. I have no idea if that's related but figured I'd throw it in there anyway.
What would be the best thing to do now? I can't really afford to call the landlord to fix it because we smuggled cats into our apartment and he may kick us out if he discovers them. Is this going to drain on it's own eventually? Will it be safe to run the dishwasher again?