There is a free bus service along the Bath Road if you have a Senior Citizens Bus Pass, otherwise I don't think it is free (I may be wrong). Take it to the Central Bus Station and then follow the signs for the Underground - there isn't another nearer the hotel. If you take the Hoppa from the hotel, get off at Terminal 2 and go to the Underground that way (bus stops 9 or 10 coming back). Do bear in mind, Rose, that Heathrow is a considerable distance away from the centre of London and it will take time to get in on the underground. I don't think Knightsbridge is on the same line as Heathrow so you would need to change and if you tried to do it in the rush hour (which is almost all the time) you'd be exhausted by the end of the day.
As for a taxi, all I can say is that when I lived in London taxis were very expensive and, given the distance, the cost would be horrendous.
Whatever you decide to do, though, have a lovely holiday.
Ann