That's my neighborhood in London. (I stay one block away.)
From Heathrow, take the Piccadilly line to Earls Court. Use the Earls Court Road exit. Cross the street directly in front of the station--there is a traffic light--turn right after crossing and walk to the corner which is about 25 feet away. (The street is Earl Court Gardens.) Turn left and walk to the end of the block. The Premier Inn is on the corner.
The tube ride from Heathrow to Earls Court will take about 40 minutes give or take a few. I would estimate you will arrive at your hotel closer to 5:30 than 7:00
DO NOT TAKE THE HEATHROW EXPRESS. IT WILL COST YOU TEN TIMES AS MUCH. AND CONSIDERING YOU WILL HAVE TO CHANGE TO THE TUBE FROM PADDINGTON TO EARLS COURT YOU WILL PROBABLY SAVE NO TIME.
From Earls Court You could take the District line to Westminster and see Westminster Abbey, the Houses of Parliament, Big Ben (covered in scaffolding) and the London Eye.
Or you could take the Piccadilly line to either Piccadilly Circus or Leicester square.
If you are arriving on a weekday please remember it will be rush hour in the late afternnon and the trains will be crowded