You might be thinking that we cannot possibly know, because there's nothing on screen about it. But I think that this is one of those mind-bending puzzles where we seemingly have no information at all but actually we can deduce enough to make some pretty shrewd estimates.
- We saw two mission time clocks, giving the relative time rates on two decks of the ship.
- We saw a city with skyscrapers, whose sizes we can estimate from the number of floors.
- We saw just after the opening titles the approximate height of a deck relative to the skyscrapers.
- We know the deck numbers.
- We saw no appreciable tidal effects.
Let's assume general relativity as in the real universe (given than the writers made a plot point out of it), normal sized humans and time lords, and Terran standard days and hours (given the whole Mondas being a twin of Earth thing). Let's also assume common Terran sizes of skyscrapers, if we cannot see all of the windows of any one in order to count them. And let's assume that what we saw of the hospital applies generally: not more than one row of windows per floor.
It seems that we know two points and the distance between them, and the relative rates of time dilation, giving us a gradient; and we also know that tidal effects are weak. We even know that the axis of the ship is perpendicular to the axis of rotation of the black hole, because its accretion disc was shown on screen as observed from the ship.
The ship is clearly under thrust, towards the black hole, and from the alignment of the decks shown just after the opening titles it is implied that artificial gravity is designed to be a product of that thrust. Let's assume a thrust of 1G, which is consistent with what appears on screen.
Can we deduce the mass of the black hole from any or all of this? Or at least bounds upon that mass? If so, what is it? If not, what information do we lack?