Being at St Pancras means you'll be right next door to King's Cross, where the Platform 9 3/4 is set up for photo ops, and there's a new Harry Potter gift shop there too.
Then the St Pancras Hotel on just the other side of the train station is STUNNINGLY gorgeous and would serve a fabulous tea.
https://www.marriott.com/hotels/hotel-photos/lonpr-st-pancras-renaissance-hotel-london/
(and where I would want to be staying if I were getting a train out of St Pancras anyway)
And then right next to that is the British Library, and their treasure room has incredible original manuscripts from everyone from Beethoven to Leonardo DaVinci and the Beatles, a Magna Carta, a load of things. So you might not want to miss out on the things that are right there where you are going anyway :)
If you were able to stay at a hotel in the area of St Pancras/King's Cross, there's a pub right across the street from the British Library which was excellent when I was there a few years back. Really excellent.
I can't tell you to make the Tower of London more important to you than the War Rooms, that's a personal preference, I think with such a short time, that the bus idea is a good one, maybe a hop on hop off tour that they run at night would be nice. Seeing all the city lights is good too.
We went on the London Eye at night, because I have a fear of heights and figured it wouldn't be as drastic to look down, it was OK, but I felt like it was taking me farther away from the sights I wanted to see, and there was no map in the pod to tell us what we were seeing. Some people found where Buckingham Palace was, but it's not something I would do again (unless it were free).
I would advise you to keep your ground transportation very simple and very fast to get to your hotel, it's a much longer trip from the airport to the city than I thought it would be, and taking several trains and such was too much for us.
Is there a reason that you're staying near the London Eye? Could you change that? It's not near any of the things you're saying you want to see. You might be better off finding something near the Tower or near the War Rooms, wherever it is that you really want to be. If you stayed near St Pancras (chains like travelodge and Premier Inn have locations near there) they'd probably hold your luggage for you while you went sightseeing, and then it would be a short luggage trip to get to the station. I'm a big advocat of keeping life very low stress while traveling, staying on the South Bank when that's not near the places you want to see might add a lot of unnecessary stress.
How old are the kids? Experienced travelers?
Hope you have fun!
-Alison