You said "easily accessible," not "cheap."

For your first stay, a lot depends on how you plan to get from Gatwick to Heathrow. My personal preference, unless I expect to need a car at Heathrow, is the airport-to-airport bus. Most of the other options go via London with multiple changes. If your budget allows, of course, there are taxis.

Unless your Heathrow flight leaves very early, in which case you might want to get there the night before, I'd head for the Gatwick Hilton (walkway to the terminal) and catch a morning bus.

If you have plenty of time the next day and want something a bit different, take the Gatwick Express into London and stay at any hotel near Victoria Station. Next day, have a real English breakfast at a restaurant, walk around Buckingham Palace and other spots in the vicinity, take a cab to Paddington and the Heathrow Express to the airport.

For staying at Heathrow, if either of your flights is from Terminal 4, you can't beat the Heathrow Hilton for convenience. (I've never gotten a room there for less than GBP 110, which may be an issue.) T4 is a 5-minute (free) train ride from T1-2-3 (they leave every 15 minutes) so if you don't have to be there at one end or the other it's not worth going there for the hotel. In that case, the Hotel Hoppa to any of the hotels on the A4 road north of the airport is fine. True, it's not free (it's not a hotel service, it's a separate business) but at GPB 2 it won't bust your budget.