score:4
As you have a landside transit visa, this is easy - yes, you can.
A friend of mine recently blogged about spending 24 hours in Heathrow (he got the wrong day for his flight), and he had no problems, and was certainly not alone.
It's worth noting, however, that SleepingInAirports gives it a poor review, and you may find it easier to grab a hotel, either one of the airport ones, or even one in town (Earl's court is only an hour away).