I've got a 13 hour layover from 8:30 AM until 9:30 PM on Friday August 15th. What do I do during the day? I've been to London three times on holiday so I've seen most of the sites.

Is there any place to rent a day room or do they have an airline First/Business class lounge that I can buy a day pass for?

My lovely wife and I plan to spend several hours at the West London Shooting School.

Last year we had a long lay over on the way to Johannesburg - but it was at Heathrow - they had a lounge area that you could pay for (pretty cheap) a place to take a shower, change, relax, take a nap, get a massage.
I will assume that Gatwick most likely has the same kind of services - Hope you have a great time - I have to wait till next year to go again

We just did the same thing. If you go to Gatwick they have an "arrivals" lounge at the Sofitel hotel. I think you may have to be on 1st class or be a platinum member for Delta to get in. It was very nice and I recommend it. It has very nice rooms for showering. It is only open to 1pm though. You can also rent a day room at the Sofitel Hotel and they were nice as well. You have to be out of the room by 630pm. Hope this helps.
Lee