Here are my suggestions for your ABRSM Grade 5 exam.
I have chosen the pieces carefully so that you may give the best performance whilst also, (where possible) purchasing books that will contain other pieces that you may wish to play or even use in other future exams.
You can download the full syllabus here.
Purcell Rondeau (from Abdelazar), out of this book.
Gershwin But Not for Me or Ned Bennett An Urge to Splurge
C. Baermann Study in Eb or Paul Harris Study in D.