It's touristy but not sure if you've done it before.
If you get a nice day, then go for a wander along the South Bank. Start at London Bridge (and if it's a Friday/Saturday) wander through Borough Market, then walk your way along the river. There are lots of pubs to stop in at, the Tate Modern and the National. It's probably my one of my favourite ways to spend an afternoon with visitors to the city.
There's an excellent company called
London Walks that do guided tours round some of the less well-known bits of town and tell you things that you might not otherwise find about.
Gourmet Burger Kitchen is good but can be slow so not necessarily great if you're in a rush. Strada for good pizza, Wagamamas for noodles, HK Diner (in Wardour St) for great Chinese. All of them will fall into your budget. Take a wander round Soho - there are lots of good cafes, restaurants, bars and little interesting shops.
I'd give the shows a miss unless you can get some half-price tickets at Leicester Square or you're really keen on seeing something.
The 360 degree bar? I think you might be talking about the bar/restaurant at the Hilton on Park Lane. I wouldn't bother. You get a better view of London from the London Eye.