I suggest you go get a battery which you know is good. It sounds to me like you may have a weak one. If you know for a fact that your battery is good, AND you can turn the flywheel (the disc into which the stainless steel driveshaft is inserted) by hand with the spark plugs out, then you should try using the starter to turn the engine over with the spark plugs out (with the button). IMPORTANT: be sure to plug the spark plug boots onto the two plug terminals to ground them while you're cranking. I believe yours are on the magneto housing (the front of the engine) Bad things can happen otherwise.
If the motor turns over freely with the plugs out, next step is to take one of the spark plug wires, put a spark plug in it, hold the plug to a source of ground (some bare metal on the motor, I usually hold it to the spark plug hole itself) and turn the motor over. Watch the spark plug tip for a blue spark. If you see a spark, your MPEM is good.
Next, try to put the plugs back in and turn it over. If it doesn't have the power to turn it over with the plugs in, it can be one of three things:
1. Bad or weak battery (try jumping it off your car batter, NOT YOUR CAR, just the battery. you should not jump start seadoos attatched to a running car. bad things may happen!)
2. Bad/loose wire (these wires have a tendency to corrode and fray inside the insulation. Usually right at the terminal. Try using jumper cables from a good battery straight to the solenoid for pos and engine block for neg, if that works, it's likely a wiring issue)
3. Bad starter (it is totally possible for your starter to be old and rusted inside, or have bad brushes/brush wires)
Good luck