Looked at the 5045 operators manual and it does not show all the things on that fuse, but lets give this a try worst case you blow another fuse.
The problem maybe the thermistor sensor (thermistors are temperature variable resistors used as sensors.). On the thermostat housing there is a sensor that is connected to a two wire harness that goes to the injection pump; this is the "cold start advance" portion of the system so it starts easier when cold by advancing the timing.
When the sensor on the thermostat housing fails it shorts out and blows the fuse. Try unplugging it, replace fuse and see if it starts and runs. The part # is RE503242, if you are quick changing it, you don't have to drain the coolant, just make sure it's not hot at the time. You will want a pan under the tractor to catch the coolant, so as not too have to clean it up.
To comply with the exhaust emissions regulation, the timing of injection pump should be around 6 to 9 degrees before TDC. These timing values do not allow proper start-up operations when engine is cold. To ease engine start-up, a cold advance system gives to the injection pump a temporary over-timing.
When coolant temperature is below 50 C (122 F), the solenoid valve is activated and opens the cold advance circuit.
Look up the part on JD parts, and it will help you locate it. I have also attached a picture of it.
URL to JD parts
John Deere - Parts Catalog