I believe Apple's battery replacement service is $129 in the US. Unfortunately people in the UK tend to get screwed on electronics prices, so, wouldn't be surprised if it's up to double that in the UK. But I would be surprised if it was more than double.
The 2011 Air is a valuable laptop that would sell for $500-$600 in the US.
If "ditching the computer altogether" means selling it for market value minus the cost of battery replacement, then you break even. Up to you at that point. If you can sell it for more than that price, you should sell. If you can't sell it for that price, then you might as well get the battery replaced.