1. Get yourself into this mindset before you enter the meeting: you must not care whether this negotiation succeeds or not. If it does, fine; if not, there are plenty of other clients out there. Hold that mental framework throughout.
2. During the meeting, postpone the money part of the discussion. Start by reviewing the job itself, deliverables, relationships, reporting structure and anything else you can think of. Make a list in advance as it will end up as part of your proposal/contract. The reason is to make sure your potential client invests time in the potential contract, creating momentum for a successful outcome which carries over into the commercials. This is exactly what all salespeople do.
3. You don't propose an exact number when the question of day-rate (or contract value) comes up. Say "I'm thinking of something in the range of £X - £Y per day." Don't worry about pitching the required range too high and don't mentally backtrack during the meeting. This is a negotiation, not a 'go no-go' single-shot duel.
4. The client will either agree, in which case result, or they will pretend to look surprised and say "We were thinking more in terms of £Z." If that's the reply, then recognise that £Z is not the client’s highest number, look disappointed and say sadly that something a bit higher than that might work, but it will need to be reviewed after six months. (Of course if it really is too low, thank the client for his time and politely exit ... and, who knows, you just might be called back!).
5. Generally, you should always point out to the client that the agreed rate will be reviewed after a certain period (six months, a year). By default the client expects that future contracts will always be at the previously-agreed rate so it’s important to nudge them out of that mindset, as well as giving you a new negotiation-point in case it turns out you sold your services too cheaply.