Which Enzymes are Responsible for the Biodegradation of Beta-endorphin?
As far as I have been able to find out, there isn't a definitive answer to this question. 

&beta; endorphin is a peptide, and it would seem that there are numerous brain peptidase enzymes that are implicated in the hydrolysis of &beta; endorphin and other peptide neurotransmitters. These include aminopeptidase N, membrane-dipeptidase A, angiotensin-converting enzyme and neutral endopeptidase (also called enkephalinase).

For an early (1984) review of this topic see here; there doesn't seem to be a more up-to-date review, but there is quite a bit of literature on the effects of inhibitors of these various enzymes.