So, time for a new plan. I've adjusted my account settings (on the Privacy tab), and I now only allow comments from "Registered users". According to the FAQ, this includes OpenID and Facebook accounts, not just LJ accounts. I've gone back to the start of the year, and I've only received 3 legitimate anonymous comments, so this shouldn't cause any major problems and hopefully it will stem the flow. If the spammers shift over to throwaway accounts then I'll start screening comments from non-friends. If it's still a problem after that then I'll restrict comments to friends only.
If anyone else is getting a similar flood of comments then you might want to apply the same settings to your account.