Sounds like your carb needs a service....if she cranks good but won't fire pretty quickly that usually indicates either a fuel problem or a spark problem......based on age alone...I'd suspect a fuel issue.
Have you tried squirting a little pre-mix into the carb and cranking her over? if she fires quickly....service your carb (and by service I mean - rebuild it with a genuine Mikuni carb kit/parts...NO AFTERMARKET kit). The ski is 15 years old, has the carb ever been serviced before? If not, it's time.
Opps.....is this a 2005 GTI RFI LE edition? If so, then you might need to look into your fuel injector?