Draino wasn't made for toilets. Using it can damage the toilet, and/or the pipe. It can also cause someone in the bathroom to be sprayed with sewage and caustic chemicals. In general, do not try to remove a toilet clog with chemicals.
Vinegar and baking soda are mild enough to avoid caustic chemical burns. If you add the vinegar slowly, you shouldn't get sewage spraying. But, I doubt that will work in a clog this severe.
When the toilet is working normally, can you see the opening for a water jet at the bottom? If you don't see a bottom jet, and there is one, that could explain why the plunger isn't working: The opening of the jet is somewhere in the trapway, and the flange on the plunger isn't long enough to cover the jet opening. This allows the jet to act as a pressure relief outlet when you use the plunger.
Your best bet is to use an toilet auger. If you use a sink auger, you might not be able to keep the end in position to be effective against the clog.
Edit, response to comment: Yes, try a beehive plunger. A sink plunger has no flange, and will not work on a toilet. A beehive plunger will fit better than a basic toilet plunger.