I'm assuming that you were trying to use the drain plug at the bottom of the engine, not where the oil is added. Draining is done most effectively when the plug on the bottom is removed and the mower is placed level again, over a catch basin and left to sit.
Oil has gotten into the combustion chamber area. The muffler (if it now contains oil) can be washed (flushed out) in a safe solvent, and allowed to dry.
When tipping a vertical shaft engine to get to the drain plug, keep the sparkplug end of the engine higher than the rest of the engine.. and follow the safety precautions.
Tipping the crankcase end up will put oil at the combustion chamber end of the engine, and seep into the combustion chamber.. and sometimes through the crankcase passage into the carburetor area. In cases where a lot of oil gets into the combustion area.. the engine might not rotate (due to a hydraulic lock), until the oil has returned to the crankcase.
If this should happen.. allow the engine to sit for an hour or so, with the sparkplug end elevated, and most of the oil will return to the crankcase.
If the air/fuel intake area has become flooded with oil, you might need to have the engine serviced.. the carburetor might need to be removed to evacuate the oil from that area.
When the oil gets into the combustion area, the sparkplug is usually soaked.. and after cleaning or replacement, the engine will smoke (for a short time) like a fog machine. This will often foul the plug again, and create a lot of carbon in the combustion area. There isn't an effective way of removing the excess carbon aside from removing the cylinder head. For an old mower, that might not matter much.. for a new one that you'd like to get years of trouble-free service from, you might want to consider having the head removed and the oil & carbon cleaned out.