tl;dr: Clear and concise alt-text goes in the alt attribute. Full description of absolutely everything in the picture plus explanation if necessary goes on the linked page with the link being placed immediately following the text.
Added: Be sure that link is meaningful, for example, "Alt text for image of the lighthouse control room" and not "Alt text for above image" since blind people will sometimes listen to a list of link text with no other context.
@jupiter_rowland ¡Ouch! Okay, so working within those limits, you could:
Toot the first toot with the image and brief alt text.
Toot a second toot with just the full image description.
Edit the first toot to provide a link to the second toot.
The issue about link text is not, to my knowledge, solvable within current Mastodon. I need to file an issue if one has not already been filed. Obviously, you can't do anything which is not supported by the Mastodon code.