The deepest mine right now is the Tau Tona mine in South Africa at 2.4 miles deep.
Air conditioning equipment is used to cool the mine from 55 °C (131 °F) down to a more tolerable 28 °C (82 °F). The rock face temperature currently reaches 60 °C (140 °F).
The article claims that
Other information that has since been given to me has raised the question of facilities possibly as deep as 12 to 14 miles.
The claim is that facilities could be placed over 5x as deep as the deepest mine.
The article suggests
my investigation suggests that depths up to three miles are feasible
That is 1/2 a mile deeper than currently existing mines.
The claims of Nazi underground facilities sounds a little different than what I could locate.
The article states
An interesting feature of the construction was the absence of any spoil. It had been carefully scattered in hills miles away.
Yet the wiki article states
signs of large excavations
Here they are described as tunnels.
The material in the article uses a reference to a book by Robert Allen. As an aside I learn that Allen was suspected of being a KGB agent.
I couldn't find much on the underground facility in Poland.
What is weird though is that the article discusses 2 facilities built during or before WWII and then has the following summary.
What can be said with certainty is that an extensive program of secret underground construction began in earnest in the years following WW II and that program continues in effect to this day.