If you're using AA miles I would look at flying directly into Dublin. Bypass London all together. You do knpw that AA flies ORD-DUB nonstop. Much easier than RDU-LHR/LGA-DUB/BFS.

If you must fly into London for some reason, then it will cost quite a bit to just spend 24 hours there and not see a whole lot. You will be really tired and the cost of getting from/to the airport will be quite high. I guess it depends on what you want to do in London. There is enough in London to spend a week, easily, much less 24 hours.