Hi,
It's 16 miles from Heathrow to Aldwych and your method of transport really depends on the baggage you'll be carrying.
If you're travelling light - then Tube/Train are the fastest. Aldwych used to have it's own Tube stop but that closed about 10 years ago. Nearest now is Covent Garden or Holborn (Piccadilly Line) which is the Tube line from Heathrow (no changes required).
Heathrow Express is faster, but drops you in Paddington, which requires 2 changes on the Tube, so any speed advantage will be lost. It's also criticised for being expensive.
Don't recommend a car as Aldwych is in the congestion charge zone and driving around London is confusing at first.
If you're laden with luggage. A black cab is the best option - door to door. If there are several of you, then the cost will reduce significantly if you're sharing it.
More info on the website I edit - #mce_temp_url#