^^ There is a Couch that goes from Hyde Park in London to Legoland. You can also get there via train from Paddington Station (or Waterloo). The train doesn't go directly to the park, so you'd have to take a Shuttle to Legoland from Windsor & Eton station (or Windsor Riverside).
I believe there is also a line that goes directly Brighten (sp?) Pier.
You can do quite a bit for $200k. You could probably do all those trips for that amount.
My family went to England (for a month) for about $20,000. We didn't stay in the fanciest hotels in Britten, but we didn't stay in Pontins either.