Can you call the company and see if they have any recommendations?
Other than that, I can't offer any good advice, except maybe to second the other replies that suggest getting the balls out of the inside of the machine.
Parenting is fun, no doubt about it. My twins dumped cooking oil and mustard on the cream colored carpet in our dining room (all I did was go pull some laundry for folding and in that time they scaled the baby gate, got the offending items and were happily playing under the table). We could never get it all out. We tried boiling water to break up some of the grease and then patting with towels. When dried, we tried cat litter to absorb the grease, then vacuumed a day later and tried baking soda, waited another day, and then tried salt. Carpet cleaners, nothing worked. We had this horrible black circle (large one) under the table for a few years. Once the boys were older we tiled. Whoever, thought that carpet in the dining room was a great idea should be slapped, LOL.
I know that has nothing to do with your situation, but just thought I'd share in the hopes that some things may not seem quite as bad after reading about what others have experienced. The hardest part for me wasn't cleaning up the oil, it was keeping my boys out of all the attempts to clean the mess...boys!
I can't help but laugh at the site of those darn balls and then the other gal with the burst down comforter, oh my gosh, it is so not funny for you gals right now, but maybe some day you can laugh about it.
Til then, know that we sympathize.