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).

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