I think that prterrel has the PERFECT response. Shoort, to the point, professional, doesn't admit too much fault, just in case this goes any further. And yes, absolutely, positively a full refund. Call it "PR".
Honestly, I think you come off a little too apologetic. I know you feel horrible, but you honestly don't have to tell her that repeatedly.
Taking an editorial eye to it:
Please accept my heartfelt apologies. This cake was not up to my standards and I am truly sorry that it did not meet your expectations.
I will be issuing a full refund. If you would like me to transfer the money, I can do so, or, if you prefer, I can send a cheque by xpresspost. Please let me know which refund method you prefer, along with the transfer information or mailing address, and I will issue the refund immediately.
I hope that this does not affect your relationship with R. She wouldn't have made the referral if she didn't believe that I could do the work.
My sincerest apologies,
Here it is, Leah.^^^ I thought you meant the de-humidifier one.