I actually had a friend this week tell me that it would be so boring for me to be in London by myself - I had to explain, "Nah, I really like me so I'll have a good time."
Amanda, what a great attitude! You really do have to like your own company to travel alone and have a good time whatever you get up to!
I hope you suggest a meet up evening with other UKYs!! Unfortunately, I will be flying to the states on the 18th.
This is what I suggest... go to your local library or bookshop and camp out in the England/Britian/UK section and just read through and find your own copy of the most current guidebook. It will serve as a lifeline and one of the best momentos of your trip when you are catching up with your journal or trying to remember the name of that market you went to. Then you can put it on your bookshelf and throughout the year it will inspire you to want to add to the library of guidebooks!!! I write in mine- checks and underlines of the places I went, star, scribbles, bus notes whatever! Anyhow, it does pay to sit and read the difference between a Lonely Planet, Rick Steves, Let's Go, Fodor's and Frommer's. Each has it's own voice and once you start reading it you will figure out which one you want to be your buddy. I'm a LP fan but will look at other guides if they are more up to date or have the info I want. Don't be afraid of sitting there for hours or making multiple visits before you commit to one book. It will be one of your best investments.
Okay... so you've got yourself a spanking new passport and one entry stamp... oh c'mon and dream a little!!! One of the best things about London is that it is just 3 hours to Paris but Eurostar via the Chunnel!! You could definitely sneak in 2-3 days in that week and another stamp in your passport.
I will admit I am not an Anglophile and the most nights I ever spent in Central London before moving here was two... so yes there is a lot to do here but you still don't want to burn out on museums and seven nights in one hostel can be a bit much especially if it turns out to be not well represented on the website. Wait until mid-end of October before booking stuff... it's off season so a couple of weeks should be sufficient.
London is also a great jumping off place for taking trips... how about Cardiff, Wales for a night or two via National Express coach... great deals within a low budget.
Or if you want to see a theatre show there are loads of websites that can get you super cheap tickets... many theatres are very small so even a cruddy seat is still half decent and the experience of seeing a West End show is a very cool way to spend an evening.
I also recommend trying a sightseeing tour. It is a great way to see a city overland. I usually ride the entire loop and remember what I want to go back and see later. It's even better if the weather is sucky because they you don't have to do all the traipsing!! Check them all out and see which you prefer. I did one that included the Thames cruise.
http://www.theoriginaltour.com/http://www.city-sightseeing.com/http://www.bigbustours.com/I can chat about independent travel on the cheap for hours so if you want any help feel free to ask me! That is one of the best things you will like when you start mingling with other backpackers... people who have been there and done that get a twinkle in their eyes when sharing experiences.