I wrote a a couple of days ago about having tried out a new way of converting MIDI files to lilypond. I posted the gist of the idea to the lilypond users' mailing list, and got some more suggestions of things to try.
The idea I liked best was that the MuseScore program has an experimental Capella import (and lilypond export), which would have let me avoid using the MIDI files as an exchange format at all. Unfortunately, in its current state, the import crashes on the capella files for Holborne. (I did report the bug on the MuseScore tracking program.)
So I tried several other programs that import MIDI and export lilypond, and the one that seems to work best for this particular purpose was the rosegarden one. I haven't finished a whole piece, but from what I've done, it looks like the work I have to do is work I couldn't reasonably expect a MIDI reading program to do for me.
The most time-consuming part is that the MIDI files for the Holborne are what lilypond calls "unfolded" repeats, and I want "volta" repeats. That is, when something is repeated, these MIDI files play it twice (which is what you want when you're using the MIDI file to practice with), whereas I want to print the music once with repeat signs around it. But otherwise, I'm just making the changes which are necessary because I want unbarred parts.