Jeremy Miles comment on your answer is best. Phone the front desk, tell them it's blocked. Go out for a few hours. Come back, no problem.
Also, know that it's probably not your fault, the toilet was probably already partially blocked from the previous occupant.
However, if you're desperate to fix this yourself there's one thing you can try which is to pour a kettle full of boiling (edit, see the comments below -- this might not be completely safe, try warm water instead) water into the toilet (hopefully there's enough spare room to do this). This is something like the manual flush method with the benefit that the boiling water will help break up the organics. That should be safe on most toilets and I've had it work for me before at home (sometimes it took two kettles).
On the other hand, if you cause damage or more mess then you're going to definitely need to let the hotel know and you'll be even more embarrassed. Trust me, hotel staff see much more embarrassing things than that all the time. Much more.
Many people I know would just strip down, cover their arms with a couple of plastic bags (you probably have laundry bags in the wardrobe) and stick their hand in there to fix it. Then jump in the shower. I wouldn't try, and it doesn't sound like you would either, but it does generally work.