Answering an email regarding recommendation letters
Oy.  [In the future, give people at least a month for this sort of thing] Here's what you can do.  (1) Be very polite and thank her a lot.  (2) Say that you would like her to write a letter and submit it when she gets the chance [perhaps late as needed].  (3) Email the grad department and say you're applying but one of your letters will be late.

Hopefully that works out at some level.  A few things could go wrong with this plan of course, but it's worth a shot.  In any event, be sure to thank the letter writer profusely before and after the fact.  Cards are nice.