The song they perform to is “We No Speak Americano” by Yolanda Be Cool. Which, while not traditional swing dance music I can understand the choice because they are trying to appeal to the general public.
What I do admire though, is Dax rising to the occasion in spite of his injury. They both put on a performance to be proud of.
In my opinion they have earned the “Spirit of Lindy” title they have listed on their website.
Update: Here’s an interview with them.