........Buy some rods and poke it?.....
Rods, but don't poke it. Fit the screw attachment things, and wind that clockwise* into the blockage and then pull. You'll bring chunks off at a time, until you finally break through. A pair of Mollgrips clipped onto your rods helps turn them (and don't forget to wear gloves, and clean your rods straight away whilst they're still wet).
*Because if you go the other way you'll unscrew the joint between two rods, and then have the fun of fishing a lost rod out of your drain.
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