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I want to keep the amp/speaker budget within £200 (~$400USD) but that's way more money than the £50-£100 I'd originally earmarked for this "simple" end of the solution! The more I look at it, the more it seems I'm faced with spending. Perhaps there's a cheap and straightforward solution I'm missing here?

These £50 speakers are very good, for an amp see if you can find a UK stockist for the Sonic Impact T-amp (US$30, PSU not included) which uses a similar integrated amplifier chip to the one in the Rio Receiver, which always punched well above its weight. Or again just see what Richer Sounds can do you, though they don't seem to have a name-brand integrated amp under £100.

Peter