"Unfortunately, they were about 40 ft into the kitchen drain, and their cable got stuck, and is now in our Drain......... !
They suggested calling a plumber and having them run a new drain line and hook it to the washer's drain line."

Here's where you get to teach the Roro Rooter guys,( I can't bring myself to call them plumbers) how to retrive a lost cable.
Tell the to rent, or bring, a Ridgid K-60 sewer machine with a retrival tip to attach to the cable.(See image) and if they still act dumb tell them Ridgid has a retrieval auger tip, Catalog# 27642--Model# T125 for sale . Their suggestion that you dig up and replace the drain line was way out of line when a simple operation would retrieve the broken cable. When they get the cable out of your drain, ( and it's their responsibility to do so since it's their fault in the first place) I'd for sure call in another plumbing company. This time with real plumbers. Good luck and let me know how you make out. Tom