remove all sparkplugs and hold full throttle and crank over several times to get water out of cylinders ! have a jump pack ready as battery will die ! use carb cleaner on the plugs each time you try to run after this , hope this has only just happened as internal damage might already started ! if all ok check the engine oil also as water normally gets in after this problem ! if oil is milky then change oil several times , run for 15min on hose each change until oil normal colour ! then you must take for a run for an hour to help dry it out !
The start stays engaged the entire time? I would guess it's a bad ignition switch -- it would be sending power to the starter (through the solenoid) regardless of if you switched it off.
Maybe it could be a bad starter, too. I'd start with the switch.