I think you are missing a trick, you should sell your services as a package offering for a set price. The set price should be based on what you know these companies are willing to pay and what you want to earn from it. For example for this current contract they are prepared to pay £30k for a certain outcome by a certain date (or sooner) and you think you can do this in 4 weeks (£10k). Agree to charge them £20k, as a whole not a day rate, for no longer than 8 weeks. Deliver it in 4 and earn £1k per day. They wont care that you delivered it early but they will be impressed that you delivered for £10k less than they were expecting.
Definitely this. There are some good Reddit threads where day rates for IT contractors (usually devs) are discussed. The bottom line tends to be that where upside for the client is massive (either potential profit or general urgency), you should just charge accordingly if you can be picky about the work.
Also the referrals comment is spot on too. Once you have a backlog satisfied customers and case studies you really shouldn't need to rely on recruitment agencies.