Your problem is that, for historical reasons that are too tedious to go into, some suburban railway

I'm working through (what I hope are) the last details for our trip. But I'm struggling to decipher the public transit system for the London area. I've been all through the National Rail site as well as a bunch of other sites suggested here, but there are still some questions I can't seem to find answers for.
Is there a site where I can type in an originating street address and a destination street address, cheap airline and hotel packages and find out how to get there by public transit? I thought I was on a site like that recently but now I can't find it.
Your problem is that, for historical reasons that are too tedious to go into, some suburban cheap airline and hotel packages railway services cheap airline and hotel packages remained cheap airline and hotel packages with companies that eventually became part of the national rail system rather than the London underground - especially south of the river and out (as you have no doubt found) to Purley.
Trains from Purley terminate at Victoria and you'd pick up the tube there, Assuming this is ISH accommodation at Regent's Park, you take the Victoria Line to Oxford Circus, then the Bakerloo Line to Regent's Park; or bus 24 direct from Victoria to Great Portland St.