Welcome to Tom's hardware Forums!
As you guessed, it is probably the switch at the end of the flat ribbon cable attached to a mini pc board at the front panel of the computer. You can try taking this sub-assembly out and look for the through hole solder joints that secure the round switch. Use a screw driver and short out the switch connections and try and start the computer. If it works fine, then rewire an auxiliary (momentary) switch to take the place of the Dell start button. It is challenging, but it can be done and worth a try.