I hate heights and I was totally OK on the London Eye (have been twice) so you might find you are all right with it. It feels very safe.
The South Bank is great for a wander (lovely book stalls, river, shops etc) and the Tate Modern is actually really good for kids. The aquarium is quite expensive but you can spend a long time and sharks etc never get old!
The Globe is honestly wonderful. I took my then 12 year old to see A Midsummer Night's Dream and she genuinely found it hilarious; it was so well done. So I would really recommend that. You can probably book something like a tour if you thought a whole play would be too much.
Chinatown is good for a cheap meal.
All the museums are worth doing. My daughter used to love the V&A as well as the NHM and Science museum. The Museum of London is really interesting.
The London Transport museum in Cov Gdn is good for younger kids.
Climbing to the top of St Paul's is great but prob not so much fun for you. It's very high.
I think if you just have a day, I'd pick a couple of things on the South Bank and have dinner in Chinatown (ten or fifteen minute walk over the river from the bit near the London Eye). Maybe do Tate Modern in the morning, walk back down the river stopping at Gabriel's Wharf for interesting shopping and a good pizza, do the aquarium and/or the London Eye and walk to Chinatown for dinner.