If you are staying for a week, then my vote is London Perfect. I've stayed with them almost 10 times. Can't be beat.
https://www.londonperfect.com
If you are staying for a shorter time or you want the hotel for all the amenities, I vote for Claridge's. They have a family room offer that is billed as 2 bedrooms and 2 bathrooms, connecting, but in my experience, along the connecting hallway is a third "Nanny" Room that holds a single twin bed. So, in reality, there are three bedrooms and two full bathrooms. Call and ask, don't make reservations online. The deal also usually includes breakfast.
Have fun!