OK you need to climb into the Wiring Diagrams now...waste of time and money to replace any more parts.
Download the 2003 Jeep KJ Service Manual here....the 2004 model is not listed but the 2003 KJ is close enough to the 2004 KJ except that the wiring colors may be different.
www.colorado4wheel.com/manuals/Jeep/KJ/
Section 8W has the Wiring Diagrams. The index section lets you find a component...in your case search for Starter Relay.
You go to the relevant page (8W-21-2)...always make sure you are looking at 3.7 Gas pages and not 2.4 L gas or Diesel KJ (Export Market).
You will see that +12 volts goes to the relay coil pin 86 from the ignition via fuse #28. The other side of the coil...pin85.. gets switched to ground via the PCM if the PCM is happy with various things...mainly that the Jeep is in Park or Neutral... or transmission TRS Assm for 45RFE transmission.
So the relay energizes and switches pin 30 through to pin 87...the pins you are jumping together. If fuse #8 40 Amps is OK...obviously it is OK as things work when you jumper the pins together...pin 87 sends +12 volts to the solenoid which activates the Starter Motor.
So from the wiring diagram you probably will find that there is no Ground on the bottom of the relay coil..pin 85.. ie. the PCM is not happy with something...probably the Park or Neutral Sensor are not supplying the expected Ground to the relay.
You can try starting it in both Park and Neutral and you can ground the pin #85 on the Starter Relay and see now if turning the ignition key turns over the engine...make sure you are in Park or Neutral or you could run somebody over!
If the engine does not turn over...you will need to place your Digital Multimeter on the DC Voltage scale and start measuring the various points ie. +12 volts on pin 86 with respect to chassis/engine and +12 volts on pin #85 with respect to the Battery Positive terminal.....this latter measurement will confirm if the PCM is not supplying ground to pin 85 as it should!
On this 2003 Logic Diagram the wire color on pin #86 is YL/BR ie. Yellow wire with thin Brown stripe and pin# 85 wire is Black/White....let me know if this is true on your 2004 KJ!
What transmission do you have?
Manual or Automatic....which automatic box?
There have been posts whereby a bad solenoid in the Auto box stops the engine from turning over!
Let me know what you find and we can take it from there.