The ski should wake up if you press the start button with no key - the guages should light up.
Attaching the DESS key should yield 2 chirps = ski recognizes the key and should crank.
With key attached, pressing the Start button should get the "click" (= solenoid is making internal contact) and the starter should then be trying to turn the engine flywheel thus pumping the cylinders, etc, creating vacuum thus pulling fuel into the cylinders through the carb(s), etc.....and if spark and compression and fuel are all in sync = BOOM inside the cylinders and the engine starts and takes over....the starter drops out and the engine sustains itself.
Sounds like you might have some electrical continuity issues.....run through all the connections.....wiggle, jiggle, open/close connectors, look for any corrosion/water, etc.....doesn't take much to impede a good electrical path. Check the connection to the electric boxes themselves, check all your grounds.
First and foremost though....you HAVE to start with a good battery.
