search for a book called "the elements of style"
it is a book not very long...but includes all the examples which may help you avoiding simple grammar mistakes in your assay.
however, it is true that the writing skill is very hard or impossible to improve significantly in short time, but if your really put efforts in these 45 days your have left, it is not impossible that you'll end up a score (3-4) which is enough for application for most graduate schools in US