Is it appropriate to use a previous (recent) letter of recommendation for a different application?
I do not think you should reuse the letter as is, for two reasons:


The letter writer did not intend for it to be reused, or else it would have been written more generically.  It's generally not reasonable to use such a letter outside of the circumstances envisioned/authorized by its author.
You would have to explain that there's no realistic way to contact the writer, and that is itself a red flag.  It looks suspicious to say "here's what Professor X would say if you could contact him, but you can't".  A letter isn't worth much if the committee couldn't in principle contact the writer to confirm or ask further question.


Certain rare cases might merit an exception, such as reusing a letter from someone who has since died.  (Where the death is a matter of public record, everybody assumes that of course the writer would have updated the letter if death hadn't intervened, and they are inclined towards flexibility because of the difficult situation.)

How hard would it really be to contact the letter writer?  If it's inconvenient but not impossible, then it could be your best option.