First, instead of waiting to achieve a respectable score you should write that you were working to achieve a score more commensurate with your actual ability. Working to recieve a respectable score just doesn't ring, and it leaves open the possibility that the presumably not respectable scores did indeed reflect you ability.
Second, opening your second sentence with two long introductory clauses weakens the point. The point is that you have been working towards an MA, and this has improved your academic ability, but those first two long clauses distracts from that point. "until I was able to achieve" is also is a weak passive construction. Instead say something like "until I achieved."
Third, "was able to occur" is also a wordy passive construction. It takes the emphasis off of the work you have done to improve your score and makes it appear as if change just happened. Instead, use a clear, active construction that focuses on your improving the score e.g. " Hard work has improved my academic ability during the last two years, and my stronger LSAT score reflects this change."
Also, to the extent that an LSAT score is relatively discreet from an academic career, I wouldn't conflate the two.
You do seem to have very valid reasons for schools making evaluations on the last score though...