I think you should base the first part of the evening around the interests of the 10 year old. Perhaps take in a show (not sure where you are flying into but, unless it's Stanstead or Luton, that's very doable), or focus on a short walking radius rather than jumping on and off buses or tubes - so Covent Garden/Trafalgar Square, or Parliament/Trafalgar Square, or Parliament/London Eye, or Tower Bridge/Southwark. Certainly in the Covent Garden area all the restaurants will be jumping at 10pm, as they will in any of the central locations.
You won't gain any time messing around with the Heathrow Express and Paddington - and don't even think of a cab from the airport as it's the middle of rush hour. I would just tube it into wherever you are staying, which should be central rather than out at Paddington. If you are arriving at LGW, then the Gatwick Express is the way to go and if at LCY, then DLR and tube.