The only way you can get broadband realistically in the UK without a landline is via Virgin cable or 3G dongle.
Virgin's prices are £21 a month without a landline for their basic 10Mbit package.
Mobile dongles, start at around £10 per month for very limited access.
On the other hand if you have very low usage requirements, you are almost certainly better off getting landline & bundled internet from someone like Talktalk for £20 a month. The latter at least you are getting a phone line too!
we got this as we are in UK fairly often and would like internet access. Think after much trawling around dh got a dongle from T-Mobile. I think it was the best deal, plus you can buy for a month or a week or just individual days, I think he found it was the only one which did this with such flexibility and small time chunks. If we are only there 1 weekend a month, even the cheapest landline package still doesn't make sence, This was the best one for our needs. It works fine as well for just browsing/emailing etc. think it would only be restrictive if downloading loads or whole films and stuff but for normal internet usage it's proving fine and not limiting for us.
"Mobile dongles, start at around £10 per month for very limited access."
Depends on whether you mean contract or PAYG. If using Three, then you can get around 15 GB per month for around 15 or 16 quid (with VAT change may have gone up). On contract that's the data allowance I get for 8 quid a month (half price deal when it was started in 2008).
PAYG options are generally more costly, but for OP, they may have already found one of the best deals that suits their needs.
For others, Three offers PAYG options where you buy dongle and have an allowance that is, for example "3 GB / 3 months" so the SIM needs to be topped up after either 3 months or 3 GB of transfers, whichever comes sooner. They also do 12 GB/ 12 months for more money, and have started offering newer 'premium' modems at yet higher cost.
On Ebay you can sometimes find the SIMs being sold at say 10 quid to provide 3 GB/ 3 months (I paid 7 or 8 pounds for one when I had used up the full 15 GB on contract).