How can I fix my Rally mower?

Sounds like your starter solenoid swith has had it.
When you said it cranked over quickly, but would not start, was your ignition actually switched on?

Try again leaving the iginition switched on, then run a temporary connection direct from positive terminal on battery to the starter motor connection (ie hot wire it). The motor should crank and should start.
If it does run as normal, then most likely your starter solenoid needs replacing. The metal contacts inside it will probably have burned out or got pitted.
I've had a Rally ride on for 12 years and had to replace mine years ago.

Steve
April 2009