Kerosene might damage your lawnmower engine, although it depends on how much is mixed with the gasoline and how long, if at all, the mower has been run in this condition. You can fix your lawnmower in an hour or two by draining it and cleaning the spark plug and carburettor thoroughly./
The first step is to remove as much of the kerosene and gasoline as possible. Some mowers will run on kerosene and simply smoke more than normal, but the burning does leave residue in the engine. Even if the mower is running and smoking, turn it off and remove the kerosene to prevent further damage.
The spark plug of your mower is quickly affected by kerosene. It causes problems with starting the mower. Clean or replace the spark plug if necessary to start your mower. If the mower will start without a spark plug, then start it without one. Once your mower starts, run it for a few minutes to allow the kerosene already in the carburettor to burn through.
If the mower still will not start, spray carburettor cleaner into the carburettor in order to clean it enough to get it to start with the new spark plug. If it still will not start, a professional will have to clean the carburettor and remove any residue in the engine that is keeping it from starting.