Yea, there's not much you can do if you made the mistake. The email wasn't rude by the way. It just seems that way because you're not getting the outcome you desire. You made a mistake and now she can't help you unfortunately. It's the same if you stuff up on booking a regular ticket - the airline MAY help you for making a mistake, but they are not ENTITLED to. You made a mistake, you clicked the wrong date. So you need to frame your tone accordingly because your negotiating position is weak.
What you CAN do is thank her and say something along the lines of "While I'm disappointed in this outcome, I understand the position you are in and I appreciate your efforts to assist me given that the initial error was mine. In future I will ensure you are engaged earlier should mishaps like this occur again" which 1) allows you to express your disappointment and 2) let her know that you're not disappointed in her but the system and 3) give her a bit of a guilt trip in a subtle way while accepting responsibility. You do NOT want to piss her off as she did nothing wrong, you did. I would even go so far as to do this in a phone call if you can rather than an email for the personal touch.
Then I suggest you make a phone call to the airline and see if they can help you out, but don't get your hopes up. See if something can be worked out, and if they don't want to give you your money back be prepared to accept other forms of restitution (e.g. credits toward future flights, etc). If you are a frequent flyer of the airline this will help out bigtime (maybe you should sign up before you make the call if it's free to do so). Do a phone call, not an email. It's a lot harder to say no to someone in person than in an email.
Good luck mate
