Well, how much do you value your sleep?

If you want to get the National Express coach, and leave 2h before departure at Gatwick, you'll need to be at Heathrow bus station at 04:05. That's a 3am alarm call.

If you want to get a taxi, you'd need to leave the hotel, I'd leave an hour, so you're talking 04:30. A 4am alarm call.

The bus will cost about £25, and a taxi about £70 (as noted above). I don't know if you're arriving at the hotel from a flight into LHR or from central London (or elsewhere).

A room at the Yotel (cabin-type hotel in the South Terminal) starts at around £50, and the Premier Inn Gatwick North (which is walkable to the North Terminal, even if it's not obvious what the route is, but also has a shuttle) has rooms from £29, and is usually around the £50 mark. The Sofitel (linked to the North Terminal) is dearer, around the £120 mark.