How can I load-balance a load balancer?
You can't, which is why load-balancing has nothing to do with availability.
Two load balancers should be enough. You will be using only one load balancer at a time. The name mywebsite should resolve to the VIP assigned to one of the load balancers.
You can do coarse HA and balancing with round robin dns by sticking it in front of your load balancers.