I will be in London for 11 hours on a Sunday night in October. Flying into LHR around 8pm. How late to businesses/eateries stay open on Sundays here? Is the airport close enough to access the metro or bus for a quick trip to a couple toursity things such as Big Ben or the Eye? We understand they may not be open, I really just want to see them to say I have. I figure with such a long layover (11hours) I should be able to at least wander for a couple hours instead of just sitting in the terminal. We are NOT booking a hotel. Advice?

In all honesty you're going to be stuck in the airport, but there's lots to do as it's a huge site.
If it was earlier in the day then you could have taken the tube into town for a looksee.
You are basically an hour outside of town by tube (metro/underground).
Problem is all the tourist stuff will be closed for the day.

Have a look round LHR: Heathrow: Shops and restaurants | Discount shopping at Heathrow

How late does the tube run? Is it too late to even roam around the city to just look and grab some fish and chips? I hate the thought of making it all the way to London, and not at least having some delicious food. And thanks for the airport info, I'll browse through it!

It's all covered on the LHR site ;)
Heathrow: Underground | London Tube to Heathrow

Thanks so much! Appreciate it :)
It's hard to understand the rest of the world doesn't live by the rules here in "Bourbon Country" where we stay open to 4 am every night or later... LOL