I agree with Puffer - Dover isn't really all that great. Yes, the white cliffs are pretty, and the castle is worth a visit, but if you have limited time and want a day trip from London there are much better options available. The Globe Theatre is great. Highly recommended. And if you don't mind standing during a performance you can get tickets for as little as £5. You can get a very nice afternoon tea next door, too. You can go down the river by ferry to Greenwich and visit the maritime museum and the observatory - stand on the Prime Meridian.
The V&A (Victoria & Albert) Museum is excellent. They have a lot of stuff on fashion too. A couple of years ago they had an entire exhibition on high heels.
The Museum of London is often overlooked, but it too is excellent - just up from St Pauls.
I wouldn't bother driving in London - it's no fun at all. Rail fares are high, yes, but if you don't mind travelling off-peak not quite so bad. You can check timetables and fares here https://www.nationalrail.co.uk
If you want a trip to the seaside - Brighton, Eastbourne or Hastings are all much nicer than Dover and within 90 minutes of London by rail. From Hastings you can easily go to the village of Battle, about 5 miles inland, and see where the Battle of Hastings and the Norman Conquest took place in 1066. The Abbey built by William the Conqueror to commemorate the dead from the battle is nice, with an excellent museum, and the walk around the old battlefield is scenic. If you're feeling really energetic and feel your boots are made for walking, you can go to Pevensey - between Eastbourne and Hastings - which is where William landed with his men in 1066 and follow in their footsteps 15 miles to Battle - a great hike, but maybe not in three-inch heels!
Anyway, I'm sure you'll have fun!