Front Panel Connector Won't Turn On Computer

Typically the wires for the case front-panel I/O have just 1-pin or 2-pin (female) housings. This allows complete flexibility in making the proper connections to the motherboard (male) header. You seem to imply that computer case came with a single 2x5 housing with one of the 10 positions blocked off (it's called a key), and the wire positions do not match up with the motherboard header assignments.

The simple solution is to reposition the pins/wires in the 2x5 housing. Each female pin should be carefully extracted from the housing by gently depressing the exposed retention tab. After all nine wires/pins are extracted, then re-insert the wires/pins back into the housing but in the needed positions. If you smashed the retention tab too far down and the pin can now be pulled out of the housing, then you will need to bend the tab back up.

Here's what the female pins look like with a 1x4-position housing: