Lots of helpful suggestions, but no-one has yet answered the specific question of what is a reasonable price for this journey.
£105 sounds surprisingly cheap for such a long distance taxi from London.
Popular London taxi firm Addison Lee quote around £250 - they're not the cheapest, but their prices are usually pretty competitive. For a regular taxi firm, £250 isn't surprising to me at all (I was expecting around £230).
Crowdsourced taxi site/app Uber, which is usually the cheapest option, quotes £104-£139 for their smallest, cheapest cars [Uber won't actually book inter-city journeys, as Ankur Banerjee points out, but their quotes give an indication of plausible prices].
£105 is cheaper than three people taking an off-peak train (£39.50 each), and London-based taxis are notoriously expensive - it can easily cost £50 just to get from one side of London zone 2 to the other.
Long-distance taxis are particularly expensive since you're not just paying for your journey, but also for the (usually empty) cab driving back to base after dropping you off. As well as the driver's time this includes double the fuel (which is expensive in the UK) and all the business they missed out on while driving back to base. If they get stuck in traffic, if you don't show up, if you're very late or if you do a runner, it's much more disruptive than for a short journey. Expect them to want at least part of the payment up front.