About 10 miles - about a 45 minute trip.
It's quite a distance - prob an hour on public transport. For public transport you'd need to get a bus and then the jubilee or metropolitan line. By taxi it's about 35 - 45 mins although could be more in traffic. You can do it - and the Milestone is an amazing base.
Get an answer from The Milestone Hotel and Residences staff and past guests.