Firstly great location, its up to you, but you have Harrods next to you, Hyde Park, a stones throw to Buckingham house, Museums at South Kensington, I would walk it. Nice/posh part of town. Easy to get to anything by Tube/taxi Bus.
Heathrow options.
Heathrow express > Paddington. Then either Taxi, or District or Circle line to High st Ken. I would do the Taxi with luggage. Approx 45 mins.
Slightly more difficult.
Piccadilly Line from Heathrow to Barons court, walk over to District line, same platform, ( you could go to Earls court but that then has Piccadilly underground and involves an Escalator). District to Earls court, swap to another district train same platforms to High St Ken. Not too bad, and cheap.
The issue is luggage but Piccadilly has luggage space. Approx 60>75 mins.
I am sure there is a coach service to the Earls court Gloucester road area, then taxi.
Or Black Cab taxi all the way, service is plentiful, more expensive but direct.
From High Street Ken, for shopping, up District or Circle to Notting hill gate, swap to Central line, For West end shopping get off at Marble Arch, Bond St, Oxford circus, or Tottenham court road, all will be Oxford Street. OR, Go opposite way to Shepherds bush, (two stops) for Westfield, open late night.
Sounds good Andrew as we will be staying there as well!

I have a shout out to Celestine to see if has connections for a van from Heathrow to the Hotel for my party of 6


