The groud wire isnt important... Until something goes wrong
In other words yeah you should re wire it.
The cheap easy way is just go and buy a extension cord the length you need and cut off the feamle end, strip the wires and use that.
I cant say ive ever taken one apart myself,
But im guessing it probally has resistors and such inside that hold a charge even after its unplugged. So unless you have a volt meter or a test light I cant really suggest you open the ballast.