London for 3 days in September

Your hotel is central. If you don't mind walking and using the Tube then you can easily do this yourself ...

Day 1
09:00 Hyde Park
11:00 Buckingham Palace
12:00 St. James's Park
12:30 Horse Guards
13:00 No 10 Downing Street
13:15 Westminster Abbey
14:30 Houses of Parliament & Big Ben
15:00 London Eye (Banqueting House or Cabinet War Rooms are alternative options)

Day 2
09:00 Trafalgar Square
10:00 #15 Red Bus to Tower Bridge
11:00 Tower of London
13:30 St. Paul's Cathedral
15:00 Tate Modern

Day 3
09:00 Piccadilly Circus
10:00 National Gallery
13:00 Covent Garden
15:00 British Museum

The Square Restaurant is very close to your hotel and boasts Two Michelin Stars.
Lunch menus are very good value at £35.00 For Two Courses or £40.00 For Three Courses.
I enjoyed a superb lunch there in 2011. http://www.squarerestaurant.com/Home.aspx

Sounds like a great schedule. I think we can do this. Do we need to purchase a London Pass or an Oyster Card before we come?

Major museums & galleries are free (except for special exhibitions).
Your only entry costs will be to St Pauls, Westminster Abbey, Tower of London and the London Eye (best pre booked). You can now pre purchase a Visitor Oyster card. I don't think a London Pass would be good value for you. https://tfl.gov.uk/travel-information/visiting-london/getting-around-london/best-ways-for-visitors-to-pay

Opinions on the London Pass:
http://www.tripadvisor.co.uk/ShowTopic-g186338-i17-k7690019-o10-The_London_Pass_Not_a_good_idea-London_England.html

2 for 1 offers:
http://www.tripadvisor.co.uk/Travel-g186338-c195600/London:United-Kingdom:2.4.1.Travelcards.Step.By.Step.html

Consider the Sky Garden instead of the London Eye... it's FREE (currently).

http://skygarden.london/

Take care with the weather webpage linked a couple of posts upthread. It is owned by the person advertising it in the post, and I see nothing on it that says where the data comes from, how safe the site is, or how accurate the site is. Buyer beware.

EDIT: the spam mentioned has now been removed

If you're looking to go to the Tower, I'd recommend you get there first thing as the queues can build in the late morning. Go straight to the Crown Jewels and see them first. Then come out and take a Yeoman Warder tour - good fun.

If you have the weather for it, I would recommend a Thames cruise. Go from Westminster towards Greenwich and enjoy the live commentary. If you get off the boat in Greenwich, you can stand on the Prime Meridian and be in both the eastern and western hemispheres at the same time.

Some restaurants I've enjoyed recently and can recommend are:

Dinner by Heston Blumenthal (so very very inventive! and not outrageously expensive. £100ish each.) Drinks are expensive at roughly £16 a glass so your mileage may vary.

Galvin at Windows. Lovely French inspired food with a pretty view over Hyde Park and super service. This one can get spendy but they often show up on groupon.

Hawksmoor (both the Seven Dials and the Air St locations) best Sunday roast in town according to many websites. Have a google. Sunday roast is £19.50 per person.

Wouldn't have wanted to be on this HoHo bus though:

http://home.bt.com/news/london-tourist-bus-has-roof-ripped-off-by-tree-91363996056379

Absolutely Rebecca! I don't normally drop £100 for a single meal, but perhaps I didn't explain well...Dinner has just been ranked as #7 in the list of the top restaurants in the world. It was an incredible experience, both on the food and the service fronts. Due to the celebrity chef aspect as well as the media exposure given to Heston and his creations, I think I might have been expecting the cost to have been even higher, so relatively, £100 didn't seem that awful to me for an absolutely top notch night out (I do think £16 is rather expensive for a shot of gin though!)

http://www.theworlds50best.com/list/1-50-winners

Since the OP is staying at a five-star hotel that's over £300 per night, I don't think they're looking for budget choices.

I am overwhelmed with all of your responses. Thank you so much!! I am trying to get through all of your suggestions which are great by the way. We are really not looking for a restaurant that is high end by something that is nice and that maybe the locals in London would go to.

I am sorry that I am not replying to each and every one of you but I hope to review all of your suggestions in the next few days and hope to respond.

Thank you again for your help.