Tom,
It's unfortunate that you didn't post earlier, as it would have given the group more time to offer suggestions, to help you work out the details.
In order to figure out "what to do", I'd suggest taking a quick trip to Edmonds to the Travel Store and pick up a copy of the England and France Guidebooks. They have LOTS of good information on the various sites that tourists typically visit in both countries, opening and closing hours, costs, ways to get around in the cities, good day trips, etc. They also have listings for accommodations in various price ranges and restaurants. I usually pack a few Guidebooks along, as they're an excellent source of reference during trips.
As you're "somewhat on a budget", Hostels are probably your best bet, although some budget Hotels will be somewhat in the same price range. I often use HI (YHA) Hostels. They have 7 properties in London but given the short time until departure your selection might be limited to those which have vacancies. Most Hostels serve breakfast and in the U.K. it will likely be "the full English Breakfast" (you might ask them to "hold" the Black Pudding!). As HI Hostels are a "chain", members receive not only a discount but also preference in booking. A membership is not expensive and you might find that useful.
The quickest and easiest method to get from London to Paris is via the EuroStar which travels under the Channel from London St. Pancras to Gare du Nord in Paris. It's a bit pricey so not sure if that will fit your budget?
You're fortunate to have a relative in Paris, as your Aunt will be able to give you lots of tips. Which area of Paris does she live?
A trip to Edmonds would also give you an opportunity to speak with the Travel Advisors at ETBD, if you have any specific questions.
Which London Airport are you arriving at? What do you want to see in Normandy?
Good luck and happy travels!
