See: https://www.bbc.co.uk/news/business-51415590
The only other thing I'd add is that Euston is easily within walking distance of St Pancras. But given all this already extraordinary cost it seems like you might as well connect St Pancras and Euston HS lines and have the odd Eurostar service run from Birmingham.
For travellers who don't know London, who have to lug around baggage and small children it's a massive pain.
It's well worth the effort to connect it. It would already be way better even if you had to switch train in one station.
Also inflation had to be taken into account when comparing projects that are decades apart.
So without knowing much about the details, it might well be that the costs are plausible to what was proposed.
The comparison of HS2 with 18 paths per hour in each direction, on an overground rail system stretching over 320 miles, running at high speed (theoretical uppper speed limit of 250mph), with the London Underground, is laughable. I could not find any references to your claim in the official documents.
The project was never about speed, but rather about increasing capacity. It was packaged and sold to the voters as shaving off some time to get from A to B, largely due to politrickery. The aim is to free up commuter, regional and freight capacity on existing lines ─ which will also have an overall impact on roads and motorway nework used by freight carriers.
https://assets.publishing.service.gov.uk/government/uploads/...
The point I was trying to make was that a comparison to HS1 without accounting for the train frequency is probably not fair.
A decade ago we build HS1, a 67 mile track between London and the Channel Tunnel, for a total of $7.6 billion ($112m / mile).
HS2 is longer but its 330 miles of tracks will cost $400m / mile.
The project costs make absolutely no sense: London is a special weird place in the UK where everything you build is twice as expensive as in the rest of the country. Considering most of HS1 was in London and most of HS2 is outside of London, this budget makes absolutely no sense.
What makes the whole thing even more insane is that the endpoints are not connected to the rest of the infrastructure. You have a 30 minute trip between HS1 and HS2. So if you're travelling from Birmingham to Paris you'll have to switch trains in London. For the money they're spending on this it's crazy that you're not getting it.