I live in London and get cars to/from the airport often. Personally, I wouldn't attempt public transport and lugging heavy suitcases around the tube after a transatlantic flight and suffering the effects of jet lag (and I'm local so I know my way around the tube) so I would recommend you make life easy and arrange a car service.
IMO the price you have been given by the hotel is extortionate and you can find much better prices booking directly with local cab companies.
I can recommend 2 companies I use often:
Exclusive Airports
Britannia Airport Cars
Both would be £70 - £80 from Heathrow Terminal 2 to the Park Plaza Westminster and you can pre-book and pre-pay online using a credit card.