You say it "just clicks when the key is turned" Is that to the run position or start? The fuel solenoid and other relays will click in the on position, the starter solenoid when turned to the start position will only click if it is bad, because the internal contacts are corroded and don't make contact. When you jumped the solenoid did you jump the large post to large post or a large post to the small post? Put your finger on the starter solenoid to see if it is clicking.You say jumping the solenoid turnd the engine over. The indicates the battery and electrical system is good for the starter to run but the wiring to the solenoid is not working..
Now if you do not have the starter solenoid clicking get out your manual and find the wires responsible for the starting circuit, grab your VOM or voltage tester and start checking the wires where the switch gets its power from, check each switch (PTO steering levers, brake, and some manufacturers also put a seat switch in the circuit) along the path to the starter solenoid. Each connection should have votage or a light. You may also work in reverse by starting at the solenoid and woring back to the key switch.
Find the problem before ordering parts, you could have the price of a new machine by using the shotgun approach and still not have the problem fixed.