Last year Emma and Claudia helped me with my planning to London for Christmas and their recommendations are spot on. I would try to at least consider going to one or more of their recommendations. As for the Christmas markets, we went to several of the markets, but really didn’t find anything interesting.
If you want the market experience, rides, ice skating, and a fun carnival atmosphere, I would recommend Winter Wonderland in Hyde Park. It is free to get in and it has most of the booths that I saw at the other markets. The one recommendation would be to go before 7:00 if you can, it can get quite crowded and if arrive before 7, it is easier to navigate and less waiting in line.
The other place that I would recommend for ice skating is the ice rink at the Natural History Museum. If your girls are up to it, you can combine ice skating with a museum visit.
I would also recommend the Gingerbread City (this year at the Somerset House):https://www.somersethouse.org.uk/whats-on/museum-architectures-gingerbread-city. We went last year and it was one of the best things we did for the holiday. The cities leading designers and architects create the city of the future made entirely of gingerbread and confections. They even have gingerbread workshops. Book this in advance, it sells out quickly.
Finally, we got a lot of our gifts for the holidays at Borough Market. Check the hours as you will want to go when the full market is open. The food is great and I got local gifts (great chocolates & sweets), jams, etc. Here is the link with the details. Https://boroughmarket.org.uk/
Have a great trip.
Sandy