London Hauptbahnhof?
Presumably this eastern HS terminus wouldn't need to have as many platforms as Euston, as I really can't see 18tph running north on two high-speed lines (36tph total!). I would imagine services to Leeds, Sheffield (area), possibly Nottingham (area) depending on the route taken (all three of these could be served by the same train, so let's say 4tph). Newcastle and Edinburgh = 2tph. Run it via Stansted and we can add another 4tph Stansted Express (with reduced turnaround times compared to other services). Beyond that, I draw a blank. 10tph should manage with, say, 8 platforms maximun. Based on the drawings for Piccadilly in the latest HS2 consultation, this would be in the region of 85-90m wide.
Since this line is fairly likely to arrive from a north-easterly direction, I would suggest a (very deep to avoid whatever's down there) underground station with one end at St Pancras and the other end at Euston. Initially this would be a terminus, but conveniently my proposed site points in exactly the right direction for a new HS-Great Western line...
I await all the reasons why this isn't feasible!