Staying the night before your flight at Heathrow is sensible.
There is an Aerotel hotel at T3. It is more budget oriented, I believe, but I have never stayed there.
There is the Hilton Garden Inn at T2/3. It is actually at T2 which is a 7-10 minute walk, underground, to T3. I've stayed here many times and is the easiest to get to T3 but is not cheap. (No bus or trains to hassle with.)
There are a bunch of hotels at T4 that involve a train to get you to T3. But it's fairly easy. (FYI--the train station for T3 actually covers T2 and T3 and is closer to T2.)
There is a Sofitel at T5, which would also involve a train to get to T3, and is probably the most expensive. But very nice. I've also stayed here a few times.
Then there are a slew of hotels around the perimeter of Heathrow that will require either a bus or a cab to get you to T3.
On your final full day in London, you could check out of your hotel, have them keep your bags, spend the day exploring, then in the evening pick up your bags and head to the airport.