Hi Shield -
if you could tell what connector your wire is plugging into I could tell you what you need. If it's the power connector and you just want to turn your pc on and off using a pushbutton, I think I know what kind of connectors you're dealing with. (it's like a little extension cable right, with the two prong male part looking just like the plug on the motherboard?) If so, 2-pin or 3-pin fan tails could extend that a little bit. Or if you need to make the wires a lot longer, you could get a couple of those washer-shaped crimp-on connectors that have a circle of metal and a hole through the middle, and bolt them together instead of soldering. Or you might be able to fit a Molex plug on the end of that small extension you have. When the wire gauge gets really small, sometimes soldering is your only option.
p.s. - if this is a one-time hack job, you might be able to solder with a decent quality butane lighter, or a propane torch if you can figure out how not to blast the wire into molten metal, if you just don't have a soldering iron and don't want to get one.