The red is then connected to the black wire from the light fixture and the red wire going to the switch. The white is connected to the white from the fixture and the white to the switch.
Hopefully I'm understanding correctly. I think I'm following ya.
The two red wires with the pigtail (where it turns to one wire) should be connected to the white switch wire. Then the red switch wire should be connected to the black wire from the fixture. The two (now one, pigtail) white wire should be connected directly to the white wire on the fixture...just that one, no switch wires included.
The light won't go off because you're not really switching it. Connecting the two original red wires, the red from the switch and the black from the light ALL together was simply making a direct connection to the light, with an extra wire in there that did nothing. That's also why it's hot down at the switch, and the white wire isn't.
Basic switching is this: (assuming new, black/white conductor service with no downstream devices, for clarity) The white wire is connected to the white on the fixture, period...thats it, the white wire is done. The black is connected to one side of the switch loop, and the return side of the loop is connected to the black wire on the fixture.
Last step, open the switch box one more time and color that white wire black with a marker or black electrical tape.
OH, btw...if that's the same switch that you shorted the circuit with the first time around, I would spend the extra $0.84 and replace it.