AA is T3. And 2 hours is too long for check-in really, make it 90 minutes and you'll be fine. If you turn up at 2 hours you'll just be in a longer queue anyway. For T3, the very nearest option is Hilton Garden Inn T2, which is 10 minutes walk from T3. Next easiest and usually a lot cheaper is HGI Hatton Cross, for which allow 20 minutes. Then a long list of hotels on Bath Road at all prices, regular Transport for London buses, 30 minutes plus or minus. More hotel information in both the hotel fora and the UK and Ireland forum.

There is a place to stay airside in T3, it's really only for those who can't enter the UK due to visa issues, though they can't check this very thoroughly. Still hotel prices around LHR are usually quite reasonable, and you have plenty of time. In fact you could even stay in London if you wanted to have some evening entertainment (etc).