After 3 years of service my Trisquel install has stopped booting and instead
gets stuck at a busybox prompt.
It looks like the disk is dying: that has little to do with Trisquel.
Using mke2fs on the drive and its partitions seeminly worked a couple
partitions
Do you realize you are formatting your partitions? Do you have a recent
backup of your data? Do you actually want to recover data (if so, using
'mke2fs' is not the way to go!)?
Until you buy a new disk you can use a live system (to work on the data you
backed up... right?). If you want to recover data, try PhotoRec (that you
can install in a live system).