To do all of the Tower that's open to the public, not reading anything but looking at everything, takes at least half a day. I think if you tried to do all on your list in a day you wouldn't enjoy them as much. Personally I'd ditch the IWM
Have to agree about the Clink and would personally include the London Dungeon & Madame Tussauds too, also to a degree IWM, however the IWM is free which might make it more attractive - worth it for the Trench & Blitz experiences and if you get there when it opens at 10am will take less than an hour. If you're serious about doing it all in a day, nothing much opens before 10am, however you could use the time beforehand to get on a tour-bus and hop on/off to get the pictures you want. Tower Bridge is definitely worth a look and has great views, you can easily do the whole tour under 2 hours. Useful to get an Oyster card for the day if you're planning to travel on public transport - kids under 16 go free but may have to show ID on the busses if they look older, will also need to travel with an adult to get free travel on the tube. Interested to hear how you get on.
You can walk to Big Ben and Buckingham Palace from where you are - go through St James' Park, which has sprung to life with some lovely flower displays. One or two pelicans flying around too.
The IWM has a James Bond exhibition on at the moment, which I'm sure will be fun. You can still spend a good 2-3 hours there.
That's probably your morning done, so head back to the river and walk to the Tate Modern. The Gourmet Pizza restaurant on Gabriel's Wharf does great pizzas.
The Temples are always nice to walk around ...but you're gonna be pushed for time if you want to squeeze the Tower in as well, especially at weekends.
A good place for Sunday lunch is the Queen Mary - it's a permanently moored boat in between Waterloo and Blackfriar's bridge. Other than wobbling around in the big wake of tiny boats (or maybe that was the beer talking), there's usually a live Jazz musician.
Other than that, go here and get 2 for 1 tickets for most attractions:
The vouchers say you need a valid train ticket for that day, but they've always worked without for me..! :)
For an evening's entertainment, a must see is Avenue Q, followed by a good nosh up in Chinatown.
I don't know if you've been to London lately, but book in advance and arrive early!