GirlWithTheMouseyHair - your ride sounds awesome. A quick bit of googling says that is roughly 60 miles a day for 7 days, is that right?
Whether your bike will be suitable really depends on its spec - whether it is a 'fast hybrid' - i.e. more or less a road bike but with flat bars, or whether it is very much a town pop to the shops bike. I would recommend that you have a bike with at least an 8 cog rear cassette and either a double or triple chain ring on the front, depending how hilly it is. If it is flat, then a double will be fine, but if there are any sharp climbs, you may be grateful for a triple ring.
Whether you want a bike with drop handlebars or whether you stick to flat bars is mostly a matter of preference. Having drop bars will mean that you can get in a more aero position, so have to work slightly less hard to overcome air resistance, it is safer if there are any fast downhill sections to go 'on the drops' and it will give you more options regarding your cycling position, so you can shift around a bit if you are feeling tired or uncomfortable.
As a minimum, I would recommend having your bike set up so that your handlebars are level with your seat, rather than being 'sit up and beg'.
If you do go for a new bike, it is worth bearing in mind that a lighter bike will be less hard work, and you soon get used to the twitchiness, but I would not go for a pure road bike if you are not used to riding one. I would go for a tourer or a cyclo-cross bike, as these have a less aggressive and therefore more comfortable position than a pure road bike, although they look very similar.
Slicker, thinner tyres are a definite advantage, whatever bike you go for. I have 32mm Borough tyres, which came with my cyclo-cross bike, which are a bit nobbly on the sides but slick in the middle, but I normally ride with 25mm slick tyres. I find that I am nearly two miles an hour faster with the slick tyres, just through the reduction in rolling resistance. Getting toughened tyres such as Armadillos really reduce the risk of puncture.
The main London to Brighton is on once a year, but they now do other London to Brighton rides, such as a night ride, and an off-road ride.