If there is an option to restrict charges to the card, it might be worth it to keep the account open. Personally, I've closed a number of cards and I didn't die. If you're not going to need credit in the near future, the hit from closing it won't last long. So, I guess I just spoke out of both sides of my mouth, didn't I? Absent annual fee, it's a coin toss. Do whatever works for you.
As to the balance, they send you a check. Piece of cake. If you wish, spend close to the number and then let them enjoy the expense of cutting you a check for anything leftover. You're mustachian, you don't need the money right away anyway.