You're getting really close. Keep in mind that hydrolysis of an ester leads to a carboxylic acid, not a ketone. Be REALLY careful with your charges. This can cost you points on a mechanism question. Keep in mind the following equilibrium,
2 H2O <=> H3O+ + OH-
At pH 7, you have all three species present, albeit the hydronium and hydroxide at 10-7 M. Each instructor is different. Some don't mind intramolecular proton transfers, while others cringe at them.
The most important considerations to keep in mind when working out such mechanisms are (1) to provide EVERY bond making and bond breaking step so there is no loss of points; and (2) to always know what the most basic and most acidic species are present in solution. You never want OH- in an acidic solution, and you never want H+ in an alkaline solution. This is why knowledge of pKa is so important.
Your prof was clever to ask you to do this at pH 7. Sounds like an exam question to me. Now, why is this compound easily hydrolyzed at pH 7? What's so special about the leaving group?