Apparently fixing a flat requires the right tools (particularly a cross wrench). When using a normal wrench, it becomes a depressing exercise in futility. After a walk that took over an hour, we were back in action with one that actually could do the job and changed the tire in about 15 minutes.
I know this sounds boring, but I can understand now why you always see gangs and characters in movies fighting with tire wrenches. The things arent useful for much else.